Physics Laboratory Manual 1

Physics Laboratory Manual I (Compiled Experiments in Mechanics)

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CONTENTS 1

Open Laboratory Guide……………………………………………………...

5

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FORM 5……………………………………………………………………….. 9

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Significant Figures (Skills Lab 1) …………………………………………..

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Experimental Errors (Skills Lab 2)………………………………………….

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Graphs and Equations (Skills Lab 3)……………………………………….

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Composition of Concurrent Forces (Required Activity)…………………..

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Coefficient of Friction (Elective Activity)……………………………………

68

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Uniform Acceleration (Required Activity)…………………………………..

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Newton’s Second Law (Required Activity)………………………………… 83

10 Projectile Motion (Elective Activity)…………………………………………

90

11 Centripetal Force (Required Activity)………………………………………

99

12 Conservation of Energy (Required Activity)……………………………….

110

13 Conservation of Momentum (Elective Activity)……………………………

119

14 Torque and Rotational Equilibrium (Elective Activity)…………………….

126

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Open Laboratory Guide (Based on the Physics Department’s Open Laboratory Guidelines) In the open laboratory system, you need not do laboratory activities during the regular schedule of the laboratory section you are enrolled in. Instead, sessions for specific laboratory activities will be scheduled, and you are given the freedom to choose the schedule to perform the activities indicated in your syllabus. Along with the freedom to do the laboratory activities on your preferred session, comes the responsibility for you to see to it that you complete all course requirements. Contingencies like typhoons, sickness or traffic problems will no longer be a concern, as you can choose to “make up” for your missed activity simply by signing up for another session. You are expected to prepare before coming to class to optimize the use of laboratory hours. 1.0 Subjects The following courses will follow the Open Laboratory system: LBYPH11, LBYPHY1 (PHYLAB1, PYENLA1 and PYCOLA1), LBYPHY12, LBYPHY2 (PHYLAB2, PYENLA2 and PYCOLA2), LBYPHYA (INTPYLA), LBYPHYD (BIOPLA1), LBYPHYE (BIOPLA2), PYMATLA, BIOPLAB (LBYPHYC) Requirements for each course are stated in the course syllabi. 2.0 Activities and Schedule You will perform around 10 laboratory activities that you may choose from a set of 12 to 15 activities. Some activities are however compulsory and these are indicated in your course syllabi. In addition, you will be required to pass the skill-building activities before you are allowed to proceed with the required activities and elective activities. Refer to your course syllabus for the list of skillbuilding, required and elective activities. For simplicity, all open lab sessions will be scheduled on the designated time and room of classes involved in the open lab system. 3.0 Procedures 3.1. Fill up the Open Laboratory Student Record Form (Form 5). The Form 5 is included in the Laboratory Manual. 3.2. Sign up for an open laboratory session. The schedule for each activity will be posted on the bulletin board outside the Physics Supply room (J409). If you want to perform one of the activities, you must sign up for a specific session. The sign-up sheets are also stationed at J409. You have the first priority to register for all the activities/sessions that are assigned on your original class schedule (i.e. the

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schedule indicated in your EAF). However, you must sign up at least 7 days before the scheduled activity otherwise, your slot will be considered open to other students. If you wish to attend an activity outside your original class schedule, the earliest that you may register for an activity is 6 days before the scheduled activity/session. The number of activities that you may perform in one week is completely upon your discretion. You must however complete the minimum requirement for the course by the end of the 12th week. You will perform the laboratory activities in groups of two or three and you may choose your own group mates. If you wish to maintain your group throughout the course, you and your group mates have to sign up for the same set of sessions. If you fail to register, or were closed out for a particular session, you may fall in line before the start of the session and will be considered a walk-in student. You will be accommodated after the initial grouping if some slots remain, and upon the discretion of the session faculty. 3.3. Read the write-up for the activity in the Laboratory Manual and answer the guide questions. 3.4. Show up for the session on time. You are expected to be punctual. You will be deemed to have forfeited your slot after the session faculty has assigned the initial groupings. You must bring the following to class: • Laboratory manual • Photocopy of the guide questions • Calculator, pen and paper • Your Form 5 and student ID If you come to class unprepared, you will be asked to leave the room and forfeit your slot. Failure to attend the specific session is subjected to penalty: • first offense : 5% deduction from the group report • second offense : 10% deduction from the group report • third offense : 20% deduction from the group report • fourth offense : 40% deduction from the group report • fifth offense : will get a score of zero for the activity 3.5. During the session After setting up the apparatus, you must have the set-up checked by the session faculty before proceeding.

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After performing the experiment, processing the data and answering the guide questions, the group must submit the worksheet (Data Sheet, graphs, computation sheets, and answers to guide questions) to the session faculty, who shall determine the reasonableness of your results. If your work is unsatisfactory, you will be asked to repeat the activity. The session faculty will grade only satisfactory worksheets. A session lasts exactly three hours (including time allotted for assessing your work). If you are unable to finish the experiment and have your report checked, you have to sign up for another session (catch-up session) to complete or repeat the work. 3.6. At the end of your session Ask the session faculty to fill up the appropriate cells in your Form 5. Keep the Form 5 until the end of the course. 3.7. If you finish early And your group feels that you have enough time to perform another activity, you may enter another session as walk-in students provided that • There are slots available in the other session • At least two of you walk in • The session faculty agrees that there is reasonable time left for you to complete the activity • You have prepared for activity and brought the necessary documents 4.0 Skills Lab Quizzes 4.1 You must first pass the Skill Building Activity/ies before you will be allowed to take the Skills Lab quiz. 4.2 Register/sign-up for the quiz session. Sign-up sheets are placed at J409. 4.3 Bring your Form 5, pen, paper, graphing paper, calculator and ID to the quiz session. 4.4 After the session faculty finishes checking your paper, ask him/her to record your score on your Form 5. 5.0 Pre-Laboratory Quizzes Quiz for laboratory activities may be taken only once. Please refer to your course syllabus for the list of laboratory activities. Ask your session faculty to record you score for the pre-lab quiz on your Form 5.

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PHYSICS DEPARTMENT College of Science De La Salle University - Manila

(Adapted from the Physics Department Form 5) Name:___________________________________________________________ COURSE: ___________________ SECTION: ___________________ ACTIVITY TITLE

DATE

TEACHER’S EVALUATION*

PRE-LAB QUIZ

WRITTEN REPORT

*All erasures should be countersigned by the session faculty

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SESSION FACULTY SIGNATURE

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EXPERIMENT NO.

Skills Lab 1

SIGNIFICANT FIGURES OBJECTIVES: To develop skills in measurement using the vernier caliper and micrometer caliper. To apply the rules for significant figures in experimental computations. To perform simple algebraic operations following the rules of significant figures. THEORY: Measurement of physical quantities is an important aspect one has to deal with in physics. It is from measurements of quantities where one deduces or confirms basic physical laws. In fact, this process of deducing or confirming conclusions from measured quantities is an underlying tenet of all the sciences – physical, behavioral or social. Indeed, measurement is a cornerstone of the scientific method. Most physical measurements involve the reading of some scale. However, the finesse of the graduation of the scale is limited and the width of the lines marking the boundaries is by no means zero. This leads the observer to estimate the last digit of the measurement. Thus the numbers resulting from measurements are to some extent uncertain. The level of uncertainty depends on the apparatus used; the skill of the observer and the number of experiment performed. The way the measured number is written or reported implies this level of uncertainty. For example in Figure 1 the length of the pencil using ruler B is between the 10 cm and the 20mm mark. It is certain that the length of the pencil is greater than 10mm and less than 20 mm. However, a portion of the length of the ruler is still unaccounted for. Thus, the observer has to estimate the value, say to around 18 mm. the last digit, which is 8, is uncertain. On the other hand, using ruler A, the reading may be 18.3 mm where the last digit 3 is an estimate. The place value of the estimate reflects the accuracy of the instrument. Ruler A has an accuracy of up to the tenth place of a millimeter (mm), whereas ruler B has an accuracy of just up to the unit’s place of a millimeter (mm).

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Ruler A (mm) 18.3 mm

Ruler B (mm) 18 mm

Figure 1: Length measurement of a pencil using two rulers with different graduation.

Significant Figures The figures that can be obtained directly the measuring instrument followed by the first estimated figure of the measurement are called significant figures. Although an estimate figure is used, this figure is still significant because it gives meaningful information (although uncertain) about the measured object. One and only one estimated or doubtful figure is retained and regarded as significant in reading a physical measurement. In measurements, each digit in the measured value is defined as significant or non-significant. Since non-zero numbers give values on the measurement, all non-zero numbers are significant. Only zeros have the possibility of being non-significant. As a rule, the number of significant figures in a measurement depends on the accuracy of the instrument used, but it is incorrect to think that the number of significant figures determines the accuracy of the measurement. It is the place value of the last significant figure to the right of the decimal point, which will determine the accuracy of the instrument used in the measurement. Rules for Determining the Number of Significant Figures: 1. Values which are either exact numbers or numbers with perfect certainty contain an infinite number of significant figures. Numbers by definition often appear in calculations. Examples are the numbers two (2) and π in the expression for the circumference of a circle (i.e., c = 2 π r). These numbers are assumed to have an unlimited number of significant figures. Exact numbers that appear in simple counting operations such as the number of trials, number of vibrations, number of dots, and defined numbers such as 100 cm in one meter, 60 seconds in one minute, 7 days a week, 12 months a year, are also assumed to have an unlimited number of significant figures. Numbers measured with perfect certainty such as 7 pencils, 10

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books, 50 students, etc. also can contain an infinite number of significant figures. 2. Non-zero digits are significant. Examples:

3.5 m 24.7 kg 9,186

(2) significant figures (3) significant figures (4) significant figures

3. Zeroes between non-zero digits are significant. Examples:

90,057 m 200.063 g 84,000.05 mm


4. Zeroes to the right of a decimal point and to the right of a non-zero digit are significant. Examples:

7.0 km 3.00 x 108 m 145.0900 g


5. Zeroes to the left of an expressed decimal point and to the right of a non-zero digit are significant. Examples:

70,000.0 s 6,500.0 g 800.0 cm


6. Zeroes to the right of the decimal point and to the left of a non-zero digit are not significant (for values without non-zero digits to the left of a decimal point). The zeros are just used to show the place-value of the non-zero digits. Examples:

0.00097 m 0.000456 kg 0.0281 s


7. Zeroes to the right of a non-zero digit but to the left of an understood decimal point are not significant. Examples:

538,000 cm 720,000 g 150 s


Rules 6 and 7 can be easily addressed if the number is expressed in scientific notation, using only significant figures in the number placed in the

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argument (before the power of 10). To illustrate, the examples in rule 6 and rule 7 are presented below in scientific notation, with the number of significant figures indicated. Values Rule #6 0.00097 0.000456 0.0281 Rule #7 538,000 720,000 150

Scientific Notation

Number of Significant Figures

9.7 x 10–4 4.56 x 10–4 2.81 x 10–2

2 3 3

5.38 x 105 7.2 x 105 1.5 x 102

3 2 2

Significant Figures and Algebraic Operations Some physical quantities are usually obtained, not by direct measurement, but by using a mathematical formula. For example, the volume of a cylinder is obtained by using the formula πr2h. The radius (r) and the height (h) of the cylinder are the quantities directly measured. The final digit in the reading of these two quantities is an estimated value. In the computation of the volume, the level of accuracy of the measurement must still be reflected in the final answer. The digits which are not significant must be dropped out continually; the answer must be rounded off to keep only the correct number of significant figures. The following rules may be used for the retention of significant figures in a computation. 1. Rounding off numbers The process of rounding off numbers to a certain number of significant figures is done so as to preserve the level of accuracy of the original measurements involved in a mathematical operation. In rounding off numbers to a certain number of significant figures, retain the number of digits specified starting from the leftmost side. If the digit next to the last retained digit is greater than 4, add 1 to the last retained digit. Otherwise, simply maintain the value of the last retained digit. Examples: Round off the following numbers to three significant figures. a. 350,892 b. 86,524 c. 7.514

⇒ 351,000 ⇒ 86,500 ⇒ 7.51

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2. Additions and Subtractions When adding or subtracting measured values, the final answer should be rounded off to the accuracy of the least accurate measurement. Examples: a.

b.

5.852 m + 3.25 m + 38.6 m _ 47.702 m

809 kg + 273.2 kg + 75.699 kg 1157.899 kg

This is the least accurate measurement. It is accurate up to the tenth of a meter.

⇒ 47.7 m

Final answer rounded off up to the tenth of a meter. This is the least accurate measurement. It is accurate up to the unit’s place of a kilogram.

⇒ 1160 kg

Final answer rounded off up to the unit’s place of a kilogram.

3. Multiplication and Division In multiplication and division, the number of significant figures in the final product or quotient equals the least number of significant figures in any of the original factors. Examples: a. 10.340 cm x 1.51 cm (5 sf) (3 sf)

=

15.6154cm2 ⇒

b. 2 π x (53.70 mm)2 (4 sf)

=

18120 mm2 (4 sf)

15.6 cm2 (3 sf)

The number 2 and π both contain an infinite number of significant figures whereas the second term 53.70 has four. Thus the least number of significant figures among the factors involved is four. In this case the number 2 and π should be rounded off to one more significant figure than the least. The constant π should be rounded off to 3.1416 since the true value of π, to ten digits is 3.141592654. This gives (2.0000)(3.1416)(2884) = 18120.7488 = 18120 mm2. The final answer is rounded off to the same number as the least number of significant figures. c. (47.213 x 12.1 cm)/0.072 s = (47.2 x 12.1 cm)/0.072 s (5 sf) (3 sf) (2 sf)

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⇒ 7.9 x 102 cm/s (2 sf)

4. Square Roots and Trigonometric Functions Round off the final answer such that it has the same number of significant figures as the measure value Examples: a. sin (37.5o) = 0.608761429 ⇒ final answer: sin (37.5o) = 0.609 ____ b. √76.5 = 8.746427842 ⇒ final answer rounded off to = 8.75

The Micrometer Caliper The micrometer caliper is an instrument used for very precise measurements of external dimensions. The object to be measured is placed between the anvil and the spindle. The thimble is then rotated to advance the spindle until the object is gripped gently between the two jaws of the caliper. The ratchet is used to tighten up the grip by the same amount each time and thus avoid using too much force. The caliper consists of a fixed main scale on the sleeve and a movable auxiliary scale on the thimble. The auxiliary scale is circular and has 50 divisions. One revolution of the thimble moves the spindle by half a millimeter. This implies that the distance between adjacent lines on the thimble corresponds to 0.01 mm.

Anvil

Spindle

Sleeve

Thimble

Reading Line

Ratchet

Figure 2: Micrometer Caliper

The main scale has 25 main divisions etched on the sleeve or barrel, which is located along the trunk of the micrometer caliper. The distance between the lines is 1.0 mm. thus the maximum reading possible is 25 mm. The lines just below the main divisions divide the upper lines such that the distance between an upper line and an adjacent lower line is 0.5 mm. How to use the micrometer caliper: 1. Check the zero position of the caliper. A properly calibrated micrometer caliper must have the main and auxiliary scales simultaneously giving a zero reading when the jaws (the anvil and the spindle) of the caliper are completely closed. In case of error, add the correction (may be either positive or negative) to every reading.

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2. Place the body to be measured between the anvil and spindle. Rotate the thimble until the object is gripped gently between the two jaws of the caliper. Turn the ratchet slowly until it clicks several times. This prevents an error due to varying degrees of tightness of the jaws. 3. Read the main scale and the circular scale. Refer to the examples below. Example #1: Main scale reading: 7.00 mm Circular scale reading: + 0.435 mm _______________________________ 7.435 mm Final reading Converted to cm:

0.7435 cm

Example #2: Main scale reading: 6.50 mm Circular scale reading: + 0.203 mm _______________________________ 6.703 mm Final reading Converted to cm:

0.6703 cm

Example #3: Main scale reading: 7.00 mm Circular scale reading: + 0.224 mm _______________________________ 7.224 mm Final reading Converted to cm:

0.7224 cm

(Figures in example 1, 2 & 3 were taken from http://www.scas.bcit.bc.ca/scas/physics/labman/m1-civ1.htm)

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The Vernier Caliper Inside calipers

Main scale

Depth gauge

vernier scale Outside calipers

Figure 3. Vernier Caliper

The vernier caliper consists of a fixed part with a main engraved scale and a movable jaw with an engraved vernier scale. The main scale is calibrated in inches on the upper part and millimeters on the lower part. The lower calibration has a maximum of 200 divisions with each division equal to one mm. The vernier scale usually has 10 major divisions. The least count of the caliper is the smallest value that can be read directly from a vernier scale. For example if the least count indicated on the caliper is 0.05 mm and its vernier scale has 20 divisions, each division corresponds to a 0.05 mm. This means that the vernier scale divides one division on the main scale into 20 subdivisions. When the jaws are closed the zero line or index of the vernier scale coincides with the zero line on the main scale. When the jaws are opened, the fraction of the main scale division that the vernier scale has moved is determined by noting which vernier divisions coincides with a main scale division. How to use the vernier caliper: The vernier caliper measures lengths, outer and inner diameters, and internal depths with the use of its outside jaws or calipers, inner calipers, and depth gauge respectively. To measure the width of a small rectangular block, open the movable jaw and place between the outside jaws the block to be measured. Close the jaws on the object and do the following steps to get the reading: 1. Observe where the zero line or index of the vernier scale falls on the main scale. For example, Fig. 4 shows the zero line of vernier scale just after the 21 mm mark of the main scale. Thus the main scale reading is 21mm. 2. Note the line on the vernier scale that coincides on the main scale. In Fig.4, the vernier division marked “1” coincides exactly with a line on the main scale.

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This division is the second from the zero line. If the least count of the vernier is 0.05 mm, this means that two divisions correspond to 0.05 mm x 2, which is equal to 0.1 mm. So the scale marked “1” in the vernier coinciding with the main scale corresponds to a 0.1 mm reading. 3. Obtain the final reading by adding the main scale reading obtained in number 1 and vernier scale reading in number 2. That is: Main scale reading: 21.0 mm Circular scale reading: + 0.1 mm _______________________________ 21.1 mm or 2.11 cm Final reading

Main scale (mm)

Vernier scale

Figure 4: Final reading is 21.1 mm

REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA 5. The Art of Experimental Physics, Daryl W. Preston and Eric R. Dietz @ 1991 John Wiley & Sons, Inc. New York 6. http://www.scas.bcit.bc.ca/scas/physics/labman/m1-civ1.htm APPARATUS/MATERIALS: Rectangular Block, 25 centavo coin, DLSU ID Card, Ruler, Vernier Caliper, Micrometer Caliper PROCEDURE A. Volume of a Rectangular Block 1. Measure the length (L), width (W), and thickness (T) of the rectangular block sing a ruler. 2. Calculate the volume (V) of the block by multiplying the length, width and height using the rules of significant figures for multiplication. (Volume of a rectangular block = L x W x H) 3. Repeat steps 1 & 2 using a vernier caliper. Tabulate the results.

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B. Volume and Surface Area of a Coin 1. Measure the diameter (D), and the thickness or height (H) of a coin using a ruler. 2. Calculate the volume (V) and the area (A) of the coin using the rules of significant figures for multiplication. (Surface area (A) = πr2; Volume of a cylinder = πr2 H where r is the radius of the cylinder.) 3. Repeat steps 1 & 2 using a vernier caliper and a micrometer caliper. Tabulate the results. C. Perimeter and Thickness of a DLSU ID Card 1. Measure the thickness (T) of three identical DLSU ID cards using a vernier caliper. Divide the reading by three to get the thickness of one ID card. 2. Repeat the above procedure using a micrometer caliper. Tabulate the results. 3. Measure the length (L), and width (W) of an ID card using a ruler. 4. Calculate the perimeter of the card by adding twice the length and twice the width using the rules of significant figures for multiplication (Perimeter of a rectangular ID card = 2L + 2W)

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Laboratory Group # & Name: Date Performed: Course Code & Section: Group Members:

__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ DATA SHEET: A. Volume of a Rectangular Block Instrument

L (cm)

W (cm)

T (cm)

V (cm3)

Ruler Vernier Caliper B. Surface Area & Volume of a Coin (25 centavo coin) Instrument

R (cm)

H (cm)

A (cm2)

V (cm3)

Ruler Vernier Caliper Micrometer Caliper C. Perimeter and Thickness of a DLSU ID Card

Instrument

Thickness of three ID cards (measured) (cm)

Thickness of one ID card (calculated) (cm)

Ruler Vernier Caliper Micrometer Caliper

L (cm)

W (cm)

Perimeter (cm)

*

*

*

*Since the maximum length that the micrometer caliper can measure is only 2.5 cm, use the value obtained using the vernier caliper. Be sure to maintain the accuracy of each instrument as reflected by your decimal places. Observe the correct number of significant figures in your calculations. SAMPLE COMPUTATIONS:

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QUESTIONS: 1. Indicate the number of significant figures in the following: ______a. 50 student’s ______d. 7.80 m ______b. 24 hours/day ______e. 100,480 cm ______c. 230 kg ______f. 0.0025 cm3 2. Perform the indicated operations for the following measured values: a. 4.0659 cm x 3.81 cm = b. 378.2 m – 56 m = c. 0.005 mm + 8.25 mm + 127.3 mm = d. 9.70 x 108 m/s ÷ 1.5 s = 3. Solve the following problems: a. A rectangular paperboard measures 8.7 cm long, 4.3 cm wide and 1.75 mm thick. Find the volume of the paperboard.

b. What is the volume of a cylinder whose radius measures 10.29 mm and has a height of 6.28 cm?

4. Specify the measuring instrument (meter stick, vernier caliper, and micrometer caliper) that is appropriate in measuring the following: __________a. radius of an ordinary ring __________b. depth of a small can __________c. thickness of a credit card __________d. diameter of a small spherical metal ball __________e. height of a table

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EXPERIMENT NO.

Skills Lab 2

EXPERIMENTAL ERRORS OBJECTIVES: Identify the types of experimental errors and its sources, and explain how these errors can be reduced. Interpret data with the use of statistical methods of dealing with errors. THEORY: Measurements of physical quantities are almost always affected by factors giving rise to variations in reading. There will always be some degree of uncertainty in the results. These variations in measurements, calculations or observations of a quantity from the true or standard value are called errors. An error that tends to make an observation too high is called a positive error and one that makes it too low a negative error. Experimental errors are generally classified as systematic and random errors. 1. Systematic Errors A systematic error is one that always produces an error of the same sign, e.g., one that would make all observations too low. Systematic errors may be due to personal, instrumental or external factors. (a) Personal Errors Personal errors may arise from a personal bias of the observer in reading an instrument, in recording an observation, or his particular method of taking data, as well as mistakes in mathematical calculations. Some specific examples include: (1) Having a bias for a particular measurement. (e.g. favoring the first measurement obtained, being prejudiced in favor of the smartest member of the group or consciously taking the lowest reading, trying to fit the measurements to some preconceived idea.) (2) Taking incorrect readings form measuring instruments caused by not looking at the scale markers at a perpendicular angle. This is also called a parallax error. For instance, the position of the water level in a graduated cylinder may appear different if viewed from above or below a line of sight perpendicular to the scale. (3) Not following the rules on significant figures. (4) Human reaction time when instantaneous measurements are necessary. Personal errors may be eliminated by observing proper caution and disregarding personal biases in taking measurements.

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(b) External Errors External errors are usually caused by external conditions such as temperature, atmospheric pressure, wind, and humidity. Temperature changes may result to expansion or contraction of measuring scales. The presence of vibration may also affect the result of sensitive experiments. Steps or corrections should be taken to reduce the effect of the above mentioned factors giving rise to systematic errors in the experiment. These errors may be reduced by improving experimental techniques, using calibrated and more accurate measuring instruments, and including correction factors in the computation when necessary. 2. Random Errors A random error is one in which positive and negative errors are equally probably. Random or erratic errors appear as variations due to a large number of unpredictable conditions and other unknown factors each of which contributes to a total error. These unknown factors or unpredictable variations in experimental situations are usually beyond the control of the observer. The unpredictable fluctuations in temperature or line voltage, and the mechanical vibrations of the experimental set-up are examples of these contributing factors. Random errors may be minimized by taking a large number of observations. One may then apply the descriptive measures of statistics to arrive at certain definite conclusions about the magnitude of the errors. There are two major classes of descriptive measures. One class measures the central tendency or location, and the other class measures the dispersion or variability among the observed values. Central tendency or location is a value around which the observations tend to cluster and which typifies their magnitude. The arithmetic mean or average, median and mode are descriptive measured under this class. Dispersion or variability is the scattering of the values of a set of observations from the average value. Two sets of data may have the same average but different variability. A small variability implies a more homogeneous data. A high variability is not desirable since this implies a lesser probability of achieving the desired outcome. Some of the statistical tools that measure the dispersion or variability among the observed values are the following: deviation, average deviation, variance, and standard deviation.

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Definition of terms 1. Arithmetic Mean (x) The arithmetic mean or average is the sum of all the observed values (xi) divided by the number of observations (N) taken. It represents the best value obtained from a series of measurements. The arithmetic mean is expressed mathematically as:

2. Deviation (d) The deviation (di) of any observation (xi) of a set of observations from the mean value (x) of the set is. d i = xi – x 3. Mean Absolute Deviation or Average Deviation (d) The mean absolute deviation (commonly called as mean deviation or average deviation) is the sum of the absolute values of the deviation divided by the number of observations. It can be thought of as the average “scattering” of measured values from the mean value. The average deviation is a measure of the dispersion of the experimental measurements about the mean (i.e., it is a measure of precision). The average deviation d is written as:

The absolute value of the deviation di which is equal to di = xi = x is just the value of di without taking into account its algebraic sign. 4. Variance (σ2) The variance (σ2) of a set of observations is the average of the squares of the deviations. This is a technique to avoid the problem of negative deviations and absolute values. It is given as:

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5. Standard Deviation (σ) The positive square root of the variance is called the standard deviation (σ). It is also called the root-mean-square deviation or simply the root-mean-square. The standard deviation is used to describe the precision of the mean of a set of measurements.

The experimental value Ex of a quantity is usually expressed as: Ex = x ± σ. This value gives us the best estimate of the quantity measured. For a normal distribution of random errors, it is found that the probability that an individual measurement will fall within one standard deviation of the mean, x ± σ, which is assumed to be the true value, is 68%; for x ± 2σ, it is 95.5%; and for x ± 3σ, it is 99.7%. This justifies our discarding any measurement “off” by more than 3σ from the arithmetic mean as a mistake (not properly) measured). They are not within the range of normal errors. 6. Numerical Error Numerical error is the difference between the experimental value and the standard value. 7. Percentage Error Percentage error is defined as the difference between the experimental value and the standard value, divided by the standard value, multiplied by 100 percent. It refers to fractional part in 100 (number of parts out of each 100) that a measured value differs from the true value. The true value is often called the standard value or the theoretical value. In symbol, Percentage Error = Experimental Value – Standard Value x 100% Standard Value 8. Percentage Difference The percentage difference is used when neither of the quantities may be taken as a “standard value”. In such cases, their average or mean value (x) may be used in place of a standard value. In symbols, Percentage Difference = xi – x x 100% -x

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If there are only two observations (e.g., x1 and x2) to compare, the percentage difference is given as: Percentage Difference = x1 – x2 x 100% x1 + x2 2

Figure 1. A Curve-track Set-up MATERIALS: Curve-track as shown in Figure 1, metal ball, carbon paper, bond paper, meter stick with cursor. PROCEDURE: 1. Place a bull’s eye level on the lip of the ramp-down and level the ramp. Drop a plumb line from the lip of the ramp down to the floor, and mark the position where the tip of the plumb line touches the floor as O. This is the reference point for all horizontal distance measurements. 2. Place the metal ball at the highest position and release from rest. Observe where it will land and mark this as C. This gives you an idea of the approximate range of the metal ball. Tape a piece of bond paper on the approximate range on the table and place a carbon paper face down on top of the paper. 3. Measure the height h, the vertical distance y, and the horizontal distance x as shown in Figure 1. 4. Starting from rest at the highest position, release the metal ball ten times.

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5. Measure the distances (xn) of each of the markings made by the meal ball as it drops on the carbon paper and bond paper from the starting point marked in #1. Record these distances as x1, x2, x3, … x10. 6. Repeats steps 4 & 5 for another 10 trials using a new bond paper, and still another for another 10 trials. 7. Compute for the individual deviations, average deviation, standard deviation and the arithmetic mean. __ 8. Compute the theoretical range which is given as xtheo = 2 √xy. Compare this with the arithmetic mean by getting the percentage error. Derivation of the Theoretical Value of x in Experimental Errors

Consider the ball shown in the diagram. If the ball is released from rest at point A, its potential energy (U) is converted to kinetic energy (K). The potential energy can be transformed to kinetic energy and vice versa but in the process of transformation the total mechanical energy (ME) of the system remains constant (neglecting friction). It can be stated that the sum of the initial potential energy and initial kinetic energy is equal to the sum of the final potential energy and final kinetic energy. In equation form: (Equation 1)

ME = (U + K)initial = (U + K)final = constant

The ball is initially at rest at point A. The kinetic energy is zero (KA = 0) since it is at rest and the potential energy is UA = mgh where m is the mass of the ball, g is acceleration due to gravity (9.8 m/s2), and h is the height of the ball relative to the reference level (point B). At point B, the potential energy is zero

30

(UB = 0) and the kinetic energy is KB = ½ mv2 where K is the kinetic energy of the bob, m is the mass of the bob, and v is the speed of the bob at point B. Applying the law of conservation of mechanical energy, the total amount of mechanical energy at the initial point (point A) is equal to the total amount of mechanical energy at the final point (point B). In symbols, MEinitial = MEfinal UA + KA = UB + KB (Equation 2)

mgh + 0 = 0 + 1 mv2 2

___ v = √2gh

⇒

Once the ball is at point B, it now moves with a velocity, which is almost horizontal. From point B to point C, the motion of the bob is now similar to that of a projectile. At point B, the velocity has zero vertical components and its horizontal component is given by: (Equation 3)

⇒

v=x t

x = vt

where x is the horizontal distance and t is the time of fall from point B to point C. To find the time t, consider the vertical component of the motion. Let y be the height of the bob from point O to point B. (Equation 4)

y = 1 gt2 2

⇒

Substituting equations (2) and (4) to equation (3) gives: ___ ⇒ x = 2√hy

(Equation 5)

If the rotational motion of the ball is to be considered, include the rotational kinetic energy in equation 2. The rotational kinetic energy (Krot) of a solid sphere with its axis through the center is Krot = ½Iω2 where I is the amount of inertia of the solid sphere and ω is the angular speed. The moment of inertia (I) of a solid sphere with the axis passing through the center is given as I = 2/5mR2 where m is the mass of the solid sphere, and r is the radius. On the other hand, the angular speed (ω) is equal to v/R.

31

With the additional rotation term for kinetic energy, equation (2) becomes: mgh + 0 = 0 + 1mv2 + 1Iω2 2 2 mgh = 1 mv2 + 1 2 mR2 2 2 5

v R

2

mgh = 1mv2 + 1mv2 2 5 (Equation 6) Substituting equations (6) and (4) to equation (3) gives:

(Equation 7) REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA 5. The Art of Experimental Physics, Daryl W. Preston and Eric R. Dietz @ 1991 John Wiley & Sons, Inc. New York

32

33


__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ DATA SHEET Trial #

x in cm

Deviation (d) in cm positive negative

d2 in cm2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Arithmetic Mean (Average):

= ________cm

34

h =_________cm y = _________cm

Average Deviation:

= ________cm

Standard Deviation:

= ________cm

__ xtheo = 2√xy = _________cm

Percentage Error: _____%

x ± σ = ______________cm QUESTIONS: 1. Classify the following as to whether they are personal, instrumental, or external errors. ____________________a. incorrect calibration of scale ____________________b. bias of observer ____________________c. expansion of scale due to temperature changes ____________________d. parallax ____________________e. pointer friction ____________________f. estimation of fractional parts of scale division ____________________g. displaced zero of scale 2. Discuss the significant of the term x ± σ in the experiment.

3. An experiment was carried out to determine the specific heat of water under standard conditions. If the experiment arrived at a value of 1.1 cal/gCo and the standard value under normal conditions is 1.0 cal/gCo, what expression should be used to compare the two, percentage error or percentage difference? Show the computation.

4. Which of the following is considered as a measure of central tendency and which is considered a measure of dispersion or variability among a given set of observations? a. arithmetic mean b. average deviation c. standard deviation

________________ ________________ ________________

35

36

EXPERIMENT NO.

Skills Lab 3

GRAPHS & EQUATIONS OBJECTIVES: To apply the rules in plotting the numerical results of an experiment. To linearize parabolic and hyperbolic graphs which will verify the actual relationship between two physical quantities. To interpret the graphs and determine the relationship between two physical quantities. Formulate an equation relating two or three quantities based on the data and the graphs. THEORY: A graphical presentation is often used as an effective tool to show explicitly how one variable varies with another. By plotting the numerical results of an experiment and observing the shape of the resulting graph, a relationship between two quantities can be established. The shape of the graph gives us a clue of the relationship of the variable involved. Some of the common ones are the following: A straight-line graph indicates linear or direct relationship between two quantities. A hyperbolic graph indicates an inverse relationship. A parabolic graph tells us of a specific kind of linear or direct relationship. The specific equation relating the two variables of the graph can only be formulated when the graph is linearized. We will see how this can be done in the succeeding discussion. A. Straight Line Graphs A.1 Linear Relationship Figure 1 shows a straight-line graph that does not pass through the origin. This is a linear graph. it shows a linear relationship between the two variables. It means that there is a first-degree relationship between the Celsius readings and the Fahrenheit readings. The general equation for a linear graph is (Equation 1)

y = mx + b

37

where m and b are constants; m is the slope of the lien and b is the y-intercept. The y-intercept of the line is the value of y when x is zero. If we take y = 68o, x = 20o, and b = 32o in graph #1, the slope can be obtained using Eq. (1): (Equation 2)

68o = m(20o) + 32o m = 1.8 or 9/5

-----------------------------------------------------------------------------------------------------------Farenheit Reading vs. Celcius Reading

Fahrenheit (oF) 32 68 104 140 176

Celsius (oC) 0 20 40 60 80 Celcius Reading (°°C)

Graph #1. Fahrenheit Reading vs. Celsius Reading -----------------------------------------------------------------------------------------------------------Substituting the value of the slope obtained in Eq. (2) to Eq. (1) and considering that the y-axis is oF and the x-axis is oC, the equation relating Fahrenheit reading and Celsius reading is therefore: o

F = (9/5)oC + 32o

(Equation 3)

We can also extrapolate values from the graph. If we extend the line downward until the temperature is 0oF, we get the corresponding value in Celsius which is 17.8 oC. By interpolation, we get values within the line such as 50oC for the corresponding Fahrenheit reading of 122oF. A.2 Direct Proportionality Figure #2 shows a straight line passing through the origin. The zero values for both variables simultaneously occur. When time is doubled the distance is also doubled. In this case, we say that the distance is directly proportional to time. In general, when two variables x and y are directly proportional to each other, the equation relating them is: (Equation 4)

y α x

38

y = kx

or

k=y x

where k is the constant of proportionality. This equation shows that the quotient of the two variables is always equal to a constant. -----------------------------------------------------------------------------------------------------------Distance vs. Time

Distance (m) 0 20 40 60 80 100 120

Time (s) 0 1 2 3 4 5 6 Time (s)

Graph #2. Distance vs. Time Graph -----------------------------------------------------------------------------------------------------------In graph #2, the physical slope represents the constant k: (Equation 5)

slope = ∆y = y2 – y1 ⇒ ∆d ∆x x2 – x1 ∆t

The physical slope is always our concern in graphical analysis. The value is independent of the choice of scales and it expresses a significant fact about the relationship between the plotted variables. For example, the slope of the distance vs. time graph represents the average speed of the object. On the other hand, the geometrical slope which is defined to be tan θ, (where θ is the angle between the straight line connecting the points and the xaxis) depends on the inclination of the line and hence, on the choice of scales. B. Parabolic Graphs In general, a parabolic graph passing through the origin can be obtained for the quantities x and y obeying the following equations: (Equation 6)

y = kx2, y = kx3, y = kx4,…., y = kxn

The relationship between x and y can be expressed as y α xn. Rewriting Eq. (6),

39

y = k(constant) xn the ratio of y and xn is a constant. To verify the actual relationship, one has to linearize the graph, i.e., plot y vs. xn, where n = 2,3,4… -----------------------------------------------------------------------------------------------------------Height (y) (m) 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

Time(t) (s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Height (y) vs. time (t)

Time (s)

Graph #3. A Parabolic Graph

Graph #3 shows a parabolic graph. From Eq. (6), the value of n determines the specific equation relating x and y. By inspection, squaring the time in the data yields a direct square relationship between height and time. Thus we say, “height is directly proportionally to the square of time.” To verify this relationship, plot height vs. square of time. The result is shown in Graph #4.

40

Height (y) (m) 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

Time(t) (s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Time squared(t2) (s2) 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

Height (y) vs. Square of Time (t2)

Time Squared (s2)

Graph #4. Linearized version of Graph #3

In general, if one quantity (y) varies directly with the square of another quantity (x2) we write, y α x2. In this case n = 2. Thus the equation that correctly expresses the relationship of height (h) and time (t) in the data is: ⇒

h = k(constant) t2

h = kt2

where the constant k represents the slope of height vs. time squared graph. C. Hyperbolic Graphs Hyperbolic graphs can be obtained for quantities obeying the following equations: (Equation 7)

y = k/x, y = k/x2, y = k/x3,…., y = kxn

A hyperbolic graph indicates an inverse relationship between two quantities i.e., y α 1/xn. The specific equation can be verified by determining the value of n. For n = 1, the equation is y = k/x

41

-----------------------------------------------------------------------------------------------------------y vs. x

(y) 200 100 67 50 40 33

(x) 1 2 3 4 5 6 x

Graph #5. A Hyperbolic Graph -----------------------------------------------------------------------------------------------------------C.1 Inverse Proportionality Graph #5 shows a hyperbolic graph. To linearize it, try n = 1 such that y = 1/x. Plotting y vs. 1/x yields a straight-line graph as shown in Graph #6. Hence y is directly proportionally to 1/x or y is inversely proportional to x. In equation form yα1 x

(Equation 8)

⇒

y=k x

⇒

or

k = xy

where k is a constant which is equal to the slope of y vs. 1/x graph.

y vs 1/x

y 200 100 67 50 40 33

x 1 2 3 4 5 6

1/x 1 0.5 0.33 0.15 0.2 0.16 1/x

Graph #6. A linearized version of Graph #5 42

C.2 Inverse Square Proportionality Sometimes, plotting y vs. 1/x will not yield a straight line but plotting y vs. 1/x will yield one. This kind of relationship is called inverse square proportionality. The variable (y) is inversely proportional to the square of x. Graph #7 illustrates such a case. 2

y vs x

y 16.7 9.4 4.2 1.4 0.8 0.5 0.4 0.3

x 0.6 0.8 1.2 2.1 2.8 3.4 3.9 4.4

x

Graph #7. A Hyperbolic Graph The linearized graph is shown in Graph #8. This can only be obtained if n = 2 such that y = k x2 y vs x2

y 16.7 9.4 4.2 1.4 0.8 0.5 0.4 0.3 0.2

x 0.6 0.8 1.2 2.1 2.8 3.4 3.9 4.4 4.8

1/x2 2.78 1.56 0.69 0.23 0.13 0.09 0.07 0.05 0.04 1/ x2

Graph #8. A linearized version of Graph #7

43

D. Method of Least Squares The method of least squares is a statistical way of determining the bestfitting curve for a given set of data. If the set of data given does not yield any of the given relationships above, then the best way to plot the results would be through the application of the method of least squares. The method of least squares usually yields a straight line whose slope and whose y-intercept can be solved by applying the following equations: The slope (m) is:

m

__ _ _ xy – x y = ____________ __ _2 2 x − x

b

__ _ _ __ x2 y − x xy = ____________ __ _2 2 x − x

(Equation 9)

The y-intercept is:

(Equation 10)

where n represents the number of samples. After determining the slope (m) and the y-intercept, the equation for the best line is determined by: y = mx + b It is important that experimental data be plotted correctly for accurate graphical interpretation. To achieve this, the rules enumerated below can be of help. For Microsoft Excel users, the software provides almost all the necessary tools. All you need to do is enter the values needed and select the appropriate command.

44

Rules for Drawing Graphs on Rectangular Coordinate Paper

1. Determination of Coordinates Determine which of the quantities to be graphed the dependent variable is and which one is the independent variable. The independent variable is the quantity, which controls or causes a change in the other quantity (dependent variable) whenever it is increased or decreased. By convention, plot the independent variable along the x-axis and dependent variable on the y-axis. 2. Labeling the axes Label each axis with the name of the quantity being plotted and its corresponding unit. Abbreviate all units in standard form. 3. Choosing the scale Choose scales that are easy to plot and read. In general, choose scales for the coordinate axes so that the curve extends over most of the graph sheet. The same scale need not be used for both axes. In many cases it is not necessary that the intersection of the two axes represent the zero values of both variables. The number should increase from left to right and from bottom to top. In cases where the values to be plotted are exceptionally large or small, rewrite the numbers in scientific notation. Place the coefficients on the coordinate scale and the multiplying factor beside the unit used. 4. Location of Points Encircle each point plotted on the graph to indicate that the value lies anywhere close to that point. Draw the curve up to the circle on one side. If several curves appear on the same sheet and the points might interfere, use squares and triangles to surround the dots of the second and third curves, respectively. 5. Drawing the curve When the points are plotted, draw a smooth line connecting the points; ignore any points that are obviously erratic. “Smooth” suggests that the line does not have to pass exactly through each point but connects the general areas of significance. If there is a clue that the quantities are linear, then a straight line representing an average value should be used. There should be more or less equal number of points above and below the line. For nonlinear curves, points should be connected with a smooth curve so that the points average around the line. For Microsoft Excel users, this procedure is automatically done by a specific command. 6. Title of the Graph At an open space near the top of the paper, state the title of the graph in the form of the dependent variable (y) vs. the independent variable (x).

45


__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ APPARATUS/MATERIALS: Graphing paper, pencil and pen, ruler, Computer with Microsoft Excel EXERCISES: 1. The following data were obtained in an experiment relating time (t) (the independent variable) to the speed (v) of an accelerating object. t(s) v(m/s)

0.5 10

1.0 15

1.5 20

2.0 25

2.5 30

3.0 35

Plot these data on rectangular coordinate paper. For those with computers, use Microsoft Excel. (a) Determine the slope of the graph (b) What physical quantity does the slope represent? (c) Determine the y-intercept of the graph. What does it represent? (d) What is the equation of the curve? 2. The heating effect of an electric in a rheostat is found to vary directly with the square of the current. What type of graph is obtained when the heat is plotted as a function of current? How could the variables be adjusted so that a linear relation would be obtained?

3. The current in a variable resistor to which a given voltage is applied is found to vary inversely with the resistance. What is the shape of the current resistance curve? How could these variables be changed in order for a straight-line graph to be obtained?

46

Do the following for exercises #4 to #8 (a) Plot the given values (y vs. x). select proper coordinate scales, label plot points, draw a smooth curve through points. (b) Linearize the graph. If necessary, compute different powers of variables and plot until you get a straight line. (c) Determine the equation of the line obtained. Indicate the value of n, k, and other constants or intercepts present in the graph. 4. The data below shows how the electric field (E) due to a point charge varies with distance (r). Distance (r) in meters Electric Field (E) in N/C

1 81

2 20.3

3 9.00

4 5.06

5 3.24

6 2.25

7 1.65

8 1.27

9 1.00

5. The following values represent a particle with an x-coordinate that varies in time. Time (t) in seconds Distance (x) in meters

0 200

1 195

2 160

3 65

4 -120

5 -425

6 -880

7 -1515

6. The following values represent the motion of a particle with a y-coordinate that varies in time. Time (t) in seconds Distance (y) in meters

0 0

1 15

2 20

3 15

4 0

5 -25

6 -60

7 -105

8 -160

7. Potential energy (Us) as a function of x-coordinate for the mass-spring system. x-coordinate (m) Us in joules

-5 375

-4 240

-3 135

-2 60

-1 15

0 0

1 15

2 60

3 135

4 240

5 375

8. The values below are unknown variables x and y with a characteristic behavior. X Y

-10 10

-1 100

1 -100

2 -50

4 -25

5 -20

10 -10

20 -5

35 -2.88

40 -2.5

9. Determine the equation, which will represent the best line for the following set of data and plot the graph of the equation.

47

Method of Least Squares X 0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9 10

y 4.6 7.1 9.5 11.5 13.7 15.9 18.6 20.9 23.5 25.4

xx 0 1 4 9 16 25 36 49 64 81

xy 0 7.1 19.1 34.5 54.8 79.8 111.6 146.3 188.0 228.6

n = _____ (number of samples) _ x=

_ y=

= _____ n

= ______ n

__ x2 =

= _____ n

__ xy =

= _____ n

The slope (m) is: __ _ _ xy – x y m = __________ = ___________ = ______ __ _2 2 x − x The y-intercept is: __ _ _ __ 2 x y − x xy b = ____________ = __________ = _____ __ _2 x2 − x The equation of the best line for the data is: 48

y = mx + b = _____ x + _____

49

50

51

52

53

54

55

56

EXPERIMENT NO.

COMPOSITION OF CONCURRENT FORCES OBJECTIVES The purpose of this experiment is to use the force table to experimentally determine the force which balances two other forces. This result is checked by adding the two forces by using their components and by graphically adding the forces. Assembly

Figure 1 Force Table Assembly

There are two ways to attach the strings to the table: The first way uses the conventional ring in the center of the table and the second way uses an anchor string through the hole in the center of the table. The advantage of the anchor string is that a higher precision can be achieved because a single knot is being centered instead of the massive ring. The anchor string keeps the masses from falling to one side when the system is not in equilibrium. NOTE: In both methods it is important to adjust the pulleys so that the strings are parallel to the top surface of the Force Table, and as close to the top surface as possible. When adjusting the pulleys, don't let the ring rest on the top surface.

57

Figure 2 Ring Method of Stringing Force Table

Ring Method See Figure 2. To use this method, screw the center post up until it stops so that it sticks up above the table. Place the ring over the post and tie one 30 cm long string to the ring for each pulley. The strings must be long enough to reach over the pulleys. Place each string over a pulley and tie a mass hanger to it. NOTE: A string can be attached to the PASCO mass hanger by wrapping the string several times (4 or 5) around the notch at the top of each mass hanger.

Figure 3 Anchor Method of Stringing Force Table

Anchor String Method See Figure 3. Cut two 60cm lengths of string and tie them together at their centers (to form an "X"). Three of the ends will reach from the center of the table over a pulley; the fourth will be threaded down through the hole in the center post to act as the anchor 58

string. Screw the center post down so it is flush with the top surface of the table. Thread the anchor string down through the hole in the center post and tie that end to one of the legs. Put each of the other strings over a pulley and tie a mass hanger on the end of each string. NOTE: A string can be attached to the PASCO mass hanger by wrapping the string several times (4 or 5) around the notch at the top of each mass hanger. Theory This experiment finds the resultant of adding two vectors by three methods: experimentally, by components, and graphically. NOTE: In all cases, the force caused by the mass hanging over the pulley is found by multiplying the mass by the acceleration due to gravity. Experimental Method Two forces are applied on the force table by hanging masses over pulleys positioned at certain angles. Then the angle and mass hung over a third pulley are adjusted until it balances the other two forces. This third force is called the equilibrant (FE ) since it is the force which establishes equilibrium. The equilibrant is not the same as the resultant (FR ). The resultant is the addition of the two forces. While the equilibrant is equal in magnitude to the resultant, it is in the opposite direction because it balances the resultant (see Figure 4). So the equilibrant is the negative of the resultant: – F E = FR = FA + FB

Figure 4 The Equilibrant Balances the Resultant

59

Figure 5 Components

Component Method Two forces are added together by adding the x- and y-components of the forces. First the two forces are broken into their x- and y-components using trigonometry: FA = Ax x + Ay y and Bx x + By y

where Ax is the x-component of vector FA and x is the unit vector in the xdirection. See Figure 5. To determine the sum of FA and FB , the components are added to get the components of the resultant FR : FR = (Ax + Bx) x + (Ay + By) y = Rx x + Ry y

To complete the analysis, the resultant force must be in the form of a magnitude and a direction (angle). So the components of the resultant (Rx and Ry ) must be combined using the Pythagorean Theorem since the components are at right angles to each other: FR = R x + R y 2

2

And using trigonometry gives the angle: tan θ =

Ry Rx

Graphical Method Two forces are added together by drawing them to scale using a ruler and protractor. The second force (FB ) is drawn with its tail to the head of the first force (FA ). The resultant (FR ) is drawn from the tail of FA to the head of FB . See Figure 6. Then the magnitude of the resultant can be measured directly from the diagram and converted to

60

the proper force using the chosen scale. The angle can also be measured using the protractor.

Figure 6 Adding Vectors Head to Tail

MATERIALS NEEDED: ME-9447 Force Table, –3 pulleys and pulley clamps, –3 mass hangers, –mass set, – string,– metric ruler, –protractor, –2 sheets of paper Setup 1.

Assemble the force table as shown in the Assembly section. Use three pulleys (two for the forces that will be added and one for the force that balances the sum of the two forces).

2.

If you are using the Ring Method, screw the center post up so that it will hold the ring in place when the masses are suspended from the two pulleys. If you are using the Anchor String Method, leave the center post so that it is flush with the top surface of the force table. Make sure the anchor string is tied to one of the legs of the force table so the anchor string will hold the strings that are attached to the masses that will be suspended from the two pulleys.

3.

Hang the following masses on two of the pulleys and clamp the pulleys at the given angles: Force A = 50 g at 30° Force B = 100 g at 120°

Procedure (Experimental Method) By trial and error, find the angle for the third pulley and the mass which must be suspended from it that will balance the forces exerted on the strings by the other two masses. The third force is called the equilibrant (FE ) since it is the force which establishes equilibrium. The equilibrant is the negative of the resultant: – F E = FR = FA + FB

61

Record the mass and angle required for the third pulley to put the system into equilibrium in Table 1. To determine whether the system is in equilibrium, use the following criteria. Ring Method of Finding Equilibrium The ring should be centered over the post when the system is in equilibrium. Screw the center post down so that it is flush with the top surface of the force table and no longer able to hold the ring in position. Pull the ring slightly to one side and let it go. Check to see that the ring returns to the center. If not, adjust the mass and/or angle of the pulley until the ring always returns to the center when pulled slightly to one side. Anchor String Method of Finding Equilibrium The knot should be centered over the hole in the middle of the center post when the system is in equilibrium. The anchor string should be slack. Adjust the pulleys downward until the strings are close to the top surface of the force table. Pull the knot slightly to one side and let it go. Check to see that the knot returns to the center. If not, adjust the mass and/or angle of the third pulley until the knot always returns to the center when pulled slightly to one side. REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA Analysis To determine theoretically what mass should be suspended from the third pulley, and at what angle, calculate the magnitude and direction of the equilibrant (FE ) by the component method and the graphical method. Component Method On a separate piece of paper, add the vector components of Force A and Force B to determine the magnitude of the equilibrant. Use trigonometry to find the direction (remember, the equilibrant is exactly opposite in direction to the resultant). Record the results in Table 1.

62


__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ Graphical Method On a separate piece of paper, construct a tail-to-head diagram of the vectors of Force A and Force B. Use a metric rule and protractor to measure the magnitude and direction of the resultant. Record the results in Table 1. Remember to record the direction of the equilibrant, which is opposite in direction to the resultant. 1. How do the theoretical values for the magnitude and direction of the equilibrant compare to the actual magnitude and direction?

Table 1 Results of the Three Methods of Vector Addition

63

64

65

66

67

EXPERIMENT NO.

Elective Activity

COEFFICIENT OF FRICTION OBJECTIVES In this lab, the Dynamics Cart will be launched over the floor using the on-board spring launcher. The cart will “decelerate” over the floor under the combined action of rolling friction and the average floor slope. To determine both the coefficient of rolling friction µ r and θ, the small angle at which the floor is inclined, two separate experiments must be done. (Recall that to determine the value of two unknowns, you must have two equations.)

Figure 5.1

Theory The cart will be launched several times in one direction, and then it will be launched several times along the same course, but in the opposite direction. For example, if the first few runs are toward the east, then the next few runs will be toward the west (See Figure 5.1). In the direction which is slightly down-slope, the acceleration of the cart is given by: And the acceleration in the direction that is slightly up-slope will be:

a1 = + g sin θ − µ r g

EQN-1 (since cos θ = 1)

Numerical values for these accelerations can be determined by measuring both the distance d that the cart rolls before stopping and the corresponding time t. Given these values, the acceleration can be determined from:

a 2 = − g sin θ − µ r g

EQN-2

Having obtained numerical values for a1 and a 2 , EQN-1 and EQN-2 can be solved

simultaneously for µ r and θ

a=

2d t2

EQN-3

Having obtained numerical values for a1 and a 2 , EQN-1 and EQN-2 can be solved

simultaneously for µ r and θ. MATERIALS NEEDED:

– Dynamics Cart

Metric tape

Stopwatch

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Procedure 1. Place the cart in its starting position and then launch it. To cock the spring plunger, push the plunger in, and then push the plunger upward slightly to allow one of the notches on the plunger bar to “catch” on the edge of the small metal bar at the top of the hole. Using a stopwatch and metric tape, determine the range d and the total time spent rolling t. Record these in Table 5.1. 2. Repeat step 1 six times for each direction and enter your results in Table 5.1. 3. Using EQN-3, compute the accelerations corresponding to your data and an average acceleration for each of the two directions. 4. Using the results of step 3, determine µ r and θ by solving for the two unknowns algebraically.

REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA 5. College Physics 5th Edition, John D. Cutnell and Kenneth W. Johnson @ 2001 John Wiley and Sons, Inc. New York 6. Physics for Scientists and Engineers (with Modern Physics) 5th Edition by Raymond A. Serway & Robert J. Beichner @2000 Saunders College Publishing, Philadelphia 7. General Physics with Bioscience Essays, Jerry B. Marion @ 1979 John Wiley & Sons, Inc.

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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ Table 5.1

Average Acceleration = ______ cm/s2

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Average Acceleration = ______ cm/s2

Data Analysis Coefficient of rolling friction = ________________ Floor Angle = ________________

Questions 1.

Can you think of another way to determine the acceleration of the cart?

2. How large is the effect of floor slope compared to that of rolling friction?

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EXPERIMENT NO.

Required Activity

UNIFORM ACCELERATION Objectives In this experiment, you will investigate how the acceleration of a cart rolling down an inclined track depends on the angle of incline. From you data, you will calculate the acceleration of an object in free-fall.

Theory A cart of mass m on an incline will roll down the incline as it is pulled by gravity. The force of gravity (mg ) is straight down as shown in Figure 6.1. The component of that is parallel to the inclined surface is mg sin θ .

Figure 6.1

To determine the acceleration, you will release the cart from rest and measure the 1 time (t) for it to travel a certain distance (d ) . Since d = at 2 , the acceleration can be 2 2d calculated as a = 2 . t A plot of a versus sinθ will be a straight line with a slope equal to the acceleration of an object in free-fall, g.

Materials Needed Track with End Stop

Cart

Pivot Clamp

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Other Required Equipment Base and support rod

Stopwatch

Graph paper

Procedure 1. Set up the track as shown in Figure 6.2 with a pivot clamp and support stand. Elevate the end of the track by about 10 cm.

Figure 6.2 Figure 7.1 Equipment Set Up

2. Set the cart on the track against the end stop and record this final position in Table 8.1. (Use the non-magnetic end of the cart so it touches the end stop.) 3. Pull the cart up to the top of the track and record the initial position where the cart will be released from rest. 4. Release the cart from rest and use the stopwatch to time how long it takes the cart to reach the end stop. The person who releases the cart should also operate the stopwatch. Repeat this measurement 10 times (with different people doing the timing). Record all the values in Table 6.1. 5. Lower the end of the track by 1 cm and repeat step 4. Use the same release position. 6. Repeat step 4 for a total of 7 angles, lowering the end of the track by 1 cm for each new angle.

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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ Table 6.1: Data Initial Release position Final Position Distance traveled (d) Height on Track Time

15 cm

14 cm

13 cm

12 cm

Trial 1

Trial 2

Trial 3

Trial 4

Trial 5

Trial 6

Trial 7

Trial 8

Trial 9

Trial 10

Average Time

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11 cm

10 cm

9 cm

Data Analysis 1. 2. 3.

Calculate the average time for each angle and record it in Table 6.1. Calculate the distance traveled, d, from the initial to the final position. Use the distance traveled and average time to calculate the acceleration for each angle and record it in Table 6.2. Table 6.2: Analysis Acceleration

Height

sin θ

15 cm

14 cm

13 cm

12 cm

11 cm

10 cm

9 cm

4.

Measure the hypotenuse of the triangle formed by the track and use this to calculate sin θ for each angle. Hypotenuse

5.

Plot acceleration versus sinθ Draw the best-fit straight line and calculate its slope. Calculate the percent difference between the slope and g = 9.8 m/s2. Slope

% Difference

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Questions 1. Does your reaction time in operating the stopwatch cause a greater percentage error at higher or lower track angles?

2. How will doubling the mass of the cart affect the results? Try it.

REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA

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EXPERIMENT NO.

Required Activity

NEWTON’S SECOND LAW Objectives The purpose is to verify Newton’s Second Law, F = ma.

Theory According to Newton’s Second Law, F = ma. F is the net force acting on the object of mass m and a is the resulting acceleration of the object. For a cart of mass m1 on a horizontal track with a string attached over a pulley to a mass

m2 (see Figure 7.1), the net force F on the entire system (cart and hanging mass) is the weight of hanging mass, F = m 2 g , assuming that friction is negligible. According to Newton’s Second Law, this net force should be equal to ma, where m is the total mass that is being accelerated, which in this case is m1 + m 2 . This experiment will check to see if m1 g is equal to (m1 + m 2 )a when friction is ignored.

To obtain the acceleration, the cart will be started from rest and the time (t) it takes for it to travel a certain distance (d) will be measured. Then since d = calculated using a =

1 2 at , the acceleration can be 2

2d (assuming a = constant) t2

MATERIALS NEEDED:

– Dynamics Cart (ME-9430) – Super Pulley with clamp – String – Stopwatch – Mass balance

– Dynamics Cart Track – Base and Support rod – Mass hanger and mass set – Wooden or metal stopping block (See Procedure Step 3)

Procedure 1.

Level the track by setting the cart on the track to see which way it rolls. Adjust the leveling feet to raise or lower the ends until the cart placed at rest on the track will not move.

2.

Use the balance to find the mass of the cart and record in Table 7.1.

3.

Attach the pulley to the end of the track as shown in Figure 7.1. Place the dynamics cart on the track and attach a string to the hole in the end of the cart and tie a mass hanger on the other end of the string. The string must be just long enough so the cart hits the stopping block before the mass hanger reaches the floor.

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4.

Pull the cart back until the mass hanger reaches the pulley. Record this position at the top of Table 7.1. This will be the release position for all the trials. Make a test run to determine how much mass is required on the mass hanger so that the cart takes about 2 seconds to complete the run. Because of reaction time, too short of a total time will cause too much error. However, if the cart moves too slowly, friction causes too much error. Record the hanging mass in Table 7.1.

Figure 7.1 Equipment Set Up 5.

Place the cart against the adjustable end stop on the pulley end of the track and record the final position of the cart in Table 7.1.

6.

Measure the time at least 5 times and record these values in Table 7.1.


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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ Table 7.1 Initial Release Position

Final Position

Total distance (d)

Cart Mass

7.

Hanging Mass

Trial 1 Time

Trial 2 Time

Trial 3 Time

Trial 4 Time

Trial 5 Time

Average Time

Increase the mass of the cart and repeat the procedure.

Data Analysis 1.

Calculate the average times and record in Table 7.1.

2.

Calculate the total distance traveled by taking the difference between the initial and final positions of the cart as given in Table 7.1.

3.

Calculate the accelerations and record in Table 7.2.

4.

For each case, calculate the total mass multiplied by the acceleration and record in Table 7.2.

5.

For each case, calculate the net force acting on the system and record in Table 7.2.

6.

Calculate the percent difference between FNET and (m1 + m 2 )a and record in Table 7.2.

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Table 7.2

Cart Mass

Acceleration

(m1 + m2 )a

FNET = m2g

% Difference

Questions 1.

Did the results of this experiment verify that F = ma?

2.

Considering frictional forces, which force would you expect to be greater: the hanging weight or the resulting total mass times acceleration? Did the results of this experiment consistently show that one was larger than the other?

3.

Why is the mass in F = ma not just equal to the mass of the cart?

4.

When calculating the force on the cart using mass times gravity, why isn’t the mass of cart included?

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EXPERIMENT NO.

Elective Activity

PROJECTILE MOTION OBJECTIVES: To study the variation of vertical displacement with horizontal displacement of a projectile. THEORY: Any object that is given an initial velocity and subsequently follows a path determined by the effect of the gravitational force acting on it and by the frictional resistance of the atmosphere is called a projectile. The path followed by a projectile is called its trajectory. The instantaneous velocity (v) of a projectile can be resolved into its vertical (vy) and horizontal (vx) components.

(Equation 1)

_______ v = √vx2 + vy2

Consider a metal ball at the highest point of a ramp similar to Figure 1. At the instant the ball leaves the ramp, its subsequent motion is determined only by gravity (neglecting air friction). Assuming the ball leaves the ramp horizontally, the initial velocity of the ball as a projectile is simply vox since it has zero vertical component (voy = 0). Using Equation 1.

(Equation 2)

_______ vinitial = √vox2 + 02 = vox

There are no horizontal forces acting on the ball. This suggests that the horizontal acceleration is zero (ax = 0). Therefore, the horizontal velocity remains the same or is constant. In equation form,

(Equation 3) (Equation 4)

vx = ∆x ∆t ax = ∆vx ⇒ ∆t

ax = 0

On the other hand, there is only one force acting on the ball along the vertical and that is the force due to gravity. The force due to gravity for small distances is approximately constant. A constant vertical force results to a constant vertical acceleration (ay = constant). This vertical acceleration is the acceleration due to gravity (g) near the earth’s surface, the magnitude of which is

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equal to 9.8 m/s2. Since the vertical acceleration ay is constant, it may be expressed as: ay = ∆vy ∆t

(Equation 5)

Therefore, the motion of a projectile is a horizontal motion with constant velocity and a vertical motion with constant acceleration. In this experiment, you will observe the motion of a metal ball released from a ramp-down by recording its position at equal time intervals. From these data, you will see how a projectile behaves and observe the horizontal and vertical motions of the projectile. The trajectory may be obtained with the use of two pieces of board, the impact board and the plotting board, which are situated perpendicular to each other as shown in Figure 1. A graphing paper is attached to the face of each board. When a metal ball is released form the ramp it hits the impact board and this is recorded in the plotting board. After marking the positions of the projectile at different intervals, the points are connected to produce the trajectory of the projectile. APPARATUS/MATERIALS: Iron stand, plotting and impact boards, curve track (ramp-down), meter stick, carbon paper, graphing paper Plotting board

Carbon paper

Ramp down Impact board

Iron stand

Graphing paper

Figure 1. The curved track with the impact board and plotting board PROCEDURE: 1. Check the curved track before starting. See to it that the portion where the metal ball will eventually leave the ramp is horizontal or level. To level that portion of the ramp-down, place a bull’s eye on the lip of the ramp-down and

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adjust the screws located on the legs of the iron stand. (This step is necessary to ensure that the projectile leaves the ramp horizontally). 2. Tape a piece of graphing paper on the impact board. Place a piece of carbon paper over the graphing paper such that the darker side of the carbon paper is facing the graphing paper. 3. Get another piece of graphing paper and this time tape it to the plotting board with the left-hand edge of the graphing paper in line with the lip of the rampdown. It is preferable that the grids on the graphing paper of the plotting board coincide with those of the impact board. This will facilitate the transfer of the marks on the impact board to the plotting board. 4. Divide the graphing paper on the plotting board into several grids. Starting from the first upper horizontal line of the graphing paper, divide the grids such that there are five square grids per subdivision. Mark each subdivision. This will serve as a timing device, the unit of time being “ter”, with one subdivision or one interval equal to one “ter”. One “ter” is the time it takes the ball to travel a distance of five square grids horizontally. 5. Set the impact board on the plotting board so that the metal ball released from the ramp will hit the impact board. The impact board must be facing the lip of the ramp-down. 6. Lift the carbon paper for trial test. Release the metal ball from various points on the ramp. Find the height wherein the ball hits the impact board even up to the last interval as you move it from one interval to the next. (For the last interval, the ball should fall close to the bottom right-hand corner of the graphing paper.) Take note of the height that gives this result. This serves as starting position for the ball all throughout the experiment. *When releasing the metal ball, be sure that it stars from rest. 7. Position the impact board at the beginning of the first interval with the carbon paper in place. 8. Release the ball from the starting point marked in step 6. Check that the impact board does not move too much when the metal ball hits it. Repeat to obtain two more dots on the impact board (for the same interval) and get the average coordinate of these three points. 9. Transfer the average point on the impact board to the plotting board. Do this by extending a horizontal line from the average point to the plotting board. 10. Move the impact board backward to the start of the next subdivision previously marked in step 4, and repeat steps 8 and 9. 11. Repeat procedure 10 until the ball no longer hits the impact board. Release from the same starting position previously defined in step 6. 12. Remove the impact board without changing the orientation of the ramp-down and the plotting board with respect to each other. Release the ball once more from the starting position and observe the path followed by the metal ball. It should move along the points marked on the plotting board. 13. Trace the path taken by the metal ball on the plotting board by connecting the points. This is the trajectory of the projectile. 14. Detach the graphing paper from the plotting board. With the leftmost solid line and the uppermost solid line as your +y and +x-axes, respectively. Measure

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the distance traveled by the metal ball both horizontally and vertically after each ter. Measuring the perpendicular distance of each point from your +x and +y-axes does this. Record the values in the data table. 15. Determine the velocity and acceleration for each interval. 16. Plot y (cm) vs. t (ter) 17. Repeat the experiment. This time, place the ramp at a different height relative to the impact board but release the ball from the same starting point. REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA

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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ DATA SHEETS: Trial I. Original height of the lip of the ramp, h = ________mm

Time t (ter)

Horizontal displacement x (mm)

Horizontal Velocity Vx (mm/ter)

Horizontal Acceleration ax (mm/ter2)

Time t (ter)

0

0

1

1

2

2

3

3

4

4

5

5

6

6

7

7 Average ax = ______mm/ter2

Vertical displacement y (mm)

Vertical Velocity Vy (mm/ter)

Vertical Acceleration ay (mm/ter2)

Average ay = ______ mm/ter2

SAMPLE COMPUTATIONS:

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Trial II. New height of the lip of the ramp, h’ = ________mm

Time t (ter)

Horizontal displacement x (mm)

Horizontal Velocity Vx (mm/ter)

Horizontal Acceleration ax (mm/ter2)

Time t (ter)

0

0

1

1

2

2

3

3

4

4

5

5

6

6

7

7 Average ax = ______mm/ter2

Vertical displacement y (mm)

Vertical Velocity Vy (mm/ter)

Vertical Acceleration ay (mm/ter2)

Average ay = ______ mm/ter2

SAMPLE COMPUTATIONS:

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EXPERIMENT NO.

Required Activity

CENTRIPETAL FORCE Objectives The purpose of this experiment is to study the effects of varying the mass of the object, the radius of the circle, and the centripetal force on an object rotating in a circular path.

Theory When an object of mass m, attached to a string of length r, is rotated in a horizontal circle, the centripetal force on the mass is given by:

F=

mv 2 = mrw 2 r

where v is the tangential velocity and ω is the angular speed (v = r ω). To measure the velocity, the time for one rotation (the period, T) is measured. Then:

v=

2πr T

and the centripetal force is given by:

F=

4π 2 mr T2

MATERIALS NEEDED

- Centripetal Force Accessory (ME-8952) - stopwatch - graph paper (2 sheets) - string

- Rotating Platform (ME-8951) - balance - mass and hanger set

REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tilley @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA 5. Pasco Scientific Manual (1998)

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Procedure Part I: Vary Radius (constant force and mass) 1.

The centripetal force and the mass of the hanging object will be held constant for this part of the experiment. Weigh the object and record its mass in Table 9.1. Hang the object from the side post and connect the string from the spring to the object. The string must pass under the pulley on the center post. See Figure 9.1.

Figure 9.1 Centripetal Force Apparatus 2.

Attach the clamp-on pulley to the end of the track nearer to the hanging object. Attach a string to the hanging object and hang a known mass over the clamp-on pulley. Record this mass in Table 9.1. This establishes the constant centripetal force.

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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ Table 9.1 Varying the Radius Radius

Period (T)

T2

Mass of the object

Mass hanging over the pulley

Slope from graph

3.

Select a radius by aligning the line on the side post with any desired position on the measuring tape. While pressing down on the side post to assure that it is vertical, tighten the thumb screw on the side post to secure its position. Record this radius in Table 9.1.

4.

The object on the side bracket must hang vertically: On the center post, adjust the spring bracket vertically until the string from which the object hangs on the side post is aligned with the vertical line on the side post.

5.

Align the indicator bracket on the center post with the orange indicator.

6.

Remove the mass that is hanging over the pulley and remove the pulley.

7.

Rotate the apparatus, increasing the speed until the orange indicator is centered in the indicator bracket on the center post. This indicates that the string supporting the hanging object is once again vertical and thus the hanging object is at the desired radius.

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8.

Maintaining this speed, use a stopwatch to time ten revolutions. Divide the time by ten and record the period in Table 9.1.

9.

Move the side post to a new radius and repeat the procedure. Do this for a total of five radii.

Analysis 1. The weight of the mass hanging over the pulley is equal to the centripetal force applied by the spring. Calculate this force by multiplying the mass hung over the pulley by “g” and record this force at the top of Table 9.2. Table 9.2 Results (Varying Radius)

Centripetal Force

Centripetal Slope from slope

Percentage Difference

2. Calculate the square of the period for each trial and record this in Table 9.1. 3. Plot the radius versus the square of the period. This will give a straight line since:

 F  r =  2 T 2  4π m  4. Draw the best-fit line through the data points and measure the slope of the line. Record the slope in Table 9.1. 5. Calculate the centripetal force from the slope and record in Table 9.2. 6. Calculate the percent difference between the two values found for the centripetal force and record in Table 9.2.

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Part II: Vary Force (constant radius and mass) The radius of rotation and the mass of the hanging object will be held constant for this part of the experiment. 1. Weigh the object and record its mass in Table 9.3. Hang the object from the side post and connect the string from the spring to the object. The string must pass under the pulley on the center post. 2. Attach the clamp-on pulley to the end of the track nearer to the hanging object. Attach a string to the hanging object and hang a known mass over the clamp-on pulley. Record this mass in Table 9.3. This determines the centripetal force. 3. Select a radius by aligning the line on the side post with any desired position on the measuring tape. While pressing down on the side post to assure that it is vertical, tighten the thumb screw on the side post to secure its position. Record this radius in Table 9.3. 4. The object on the side bracket must hang vertically: On the center post, adjust the spring bracket vertically until the string from which the object hangs on the side post is aligned with the vertical line on the side post. 5. Align the indicator bracket on the center post with the orange indicator. 6. Remove the mass that is hanging over the pulley and remove the pulley. 7. Rotate the apparatus, increasing the speed until the orange indicator is centered in the indicator bracket on the center post. This indicates that the string supporting the hanging object is once again vertical and thus the hanging object is at the desired radius. 8. Maintaining this speed, use a stopwatch to time ten revolutions. Divide the time by ten and record the period in Table 9.3. 9. To vary the centripetal force, clamp the pulley to the track again and hang a different mass over the pulley. Keep the radius constant and repeat the procedure from Step #4. Do this for a total of five different forces.

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Table 9.3 Varying the centripetal Force

Mass of the object

Radius

Slope from Graph

Mass over pulley

Centripetal Force = mg

Period (T)

1 T2

Analysis 1.

The weight of the mass hanging over the pulley is equal to the centripetal force applied by the spring. Calculate this force for each trial by multiplying the mass hung over the pulley by “g” and record the results in Table 9.3.

2.

Calculate the inverse of the square of the period for each trial and record this in Table 9.3.

3.

Plot the centripetal force versus the inverse square of the period. This will give a straight line since:

F=

4π 2 mr T2

4.

Draw the best-fit line through the data points and measure the slope of the line. Record the slope in Table 9.3.

5.

Calculate the mass of the object from the slope and record in Table 9.4.

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Table 9.4 Results (varying the centripetal force)

Mass of object (from scale)

Mass of object (from slope)

Percentage Difference

6.

Calculate the percent difference between the two values found for the mass of the object and record in Table 9.4.

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EXPERIMENT NO.

Required Activity

CONSERVATION OF ENERGY Purpose The purpose is to examine spring potential energy and gravitational potential energy and to show how energy is conserved.

Theory The potential energy of a spring compressed a distance x from equilibrium is given by

PE =

1 2 kx , where k is the spring constant. According to Hooke’s Law, the force exerted by the 2

spring is proportional to the distance the spring is compressed or stretched, F = kx, where k is the proportionality constant. Thus the spring constant can be experimentally determined by applying different forces to stretch or compress the spring different distances. When the force is plotted versus distance, the slope of the resulting straight line is equal to k. The gravitational potential energy gained by a cart as it climbs an incline is given by potential energy = mgh, where m is the mass of the cart, g is the acceleration due to gravity, and h is the vertical height the cart is raised. In terms of the distance, d, along the incline, the height is given by h = d sinθ. If energy is conserved, the potential energy in the compressed spring will be completely converted into gravitational potential energy. MATERIALS NEEDED:

– Dynamics Cart with Mass (ME-9430) – Super Pulley with clamp – Base and Support rod (ME-9355) – String – Mass balance

– Dynamics Cart Track – Meter stick – Mass hanger and mass set (several kilograms) – Graph paper

Procedure 1. Level the track by setting the cart on the track to see which way it rolls. Adjust the leveling feet to raise or lower the ends until the cart placed at rest on the track will not move. 2. Use the balance to find the mass of the cart. Record this value in Table 9.2.

Determining the Spring Constant 3. Set the cart on the track with the spring plunger against the stopping block as shown in Figure 9.1. Attach a string to the cart and attach the other end to a mass hanger, passing the string over the pulley. 4. Record the cart’s position in Table 9.1.

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5. Add mass to the mass hanger and record the new position. Repeat this for a total of 5 different masses.

Figure 9.1 Equipment Set Up


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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ Table 9.1

Added Mass

Position

Displacement from Equilibrium

Force (mg)

Potential Energy 6. Remove the leveling feet. 7. Remove the string from the cart and cock the spring plunger to its maximum compression position. Place the cart against the end stop. Measure the distance the spring plunger is compressed and record this value in Table 9.2. 8. Incline the track and measure its height and hypotenuse (see Figure 9.2) to determine the angle of the track.

 height   angle = arcsin   hypotenuse  Record the angle in Table 9.2.

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9. Record the initial position of the cart in Table 9.2 10. Release the plunger by tapping it with a stick and record the distance the cart goes up the track. Repeat this five times. Record the maximum distance the cart went in Table 9.2. 11. Change the angle of inclination and repeat the measurements. 12. Add mass to the cart and repeat the measurements. Table 9.2

Distance traveled by the cart (d)

Angle

Mass

Trial 1

Trial 2

Trial 3

Distance spring is compressed (x)

Initial Position of the cart

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Trial 4

Trial 5

Max

h= dsinθ

Data Analysis 1. Using the data in Table 9.1, plot force versus displacement. Draw the best-fit straight line through the data points and determine the slope of the line. The slope is equal to the effective spring constant, k. k = ____________ 2. Calculate the spring potential energy and record in Table 9.3. 3. Calculate the gravitational potential energy for each case and record in Table 9.3. 4. Calculate the percent difference between the spring potential energy and the gravitational potential energy. Table 9.3

Angle /Mass

1 2

 

Gravitational PE (mgh)

Spring PE  kx 2 

% Difference

Questions 1. Which of the potential energies was larger? Where did this “lost” energy go?

2. When the mass of the cart was doubled, why did the gravitational potential energy remain about the same?

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EXPERIMENT NO.

Elective Activity

CONSERVATION OF MOMENTUM Experiment 11.1: Conservation of Momentum in Explosions Objectives The purpose of this experiment is to demonstrate conservation of momentum with two carts pushing away from each other.

Theory When two carts push away from each other (and there is no net force on the system), the total momentum is conserved. If the system is initially at rest, the final momentum of the two carts must be equal in magnitude and opposite in direction to each other so the resulting total momentum of the system is zero:

p = m1v1 + m2 v 2 = 0 Therefore, the ratio of the final speeds of the carts is equal to the ratio of the masses of the carts.

v1 m1 = v2 m2 To simplify this experiment, the starting point for the carts at rest is chosen so that the two carts will reach the ends of the track simultaneously. The speed, which is the distance divided by the time, can be determined by measuring the distance traveled since the time traveled by each cart is the same.

∆x1 v1 ∆x = ∆t = 1 v 2 ∆x 2 ∆x 2 ∆t Thus the ratio of the distances is equal to the ratio of the masses:

∆x1 m1 = ∆x 2 m2 MATERIALS NEEDED: Track with Feet and End Stops Collision Cart

Plunger Cart Cart Masses

Other Required Equipment

Mass set

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Procedure 1.

Install the feet on the track and level it. Install one end stop at each end with the magnetic sides facing away from the carts.

Figure 11.1

2.

For each of the cases in Table 1.1, place the two carts against each other with the plunger of one cart pushed completely in and latched in its maximum position (see Figure 11.1).

3.

Tap the plunger release button with a short stick and watch the two carts move to the ends of the track. Experiment with different starting positions until the two carts reach the ends of the track at the same time. Measure the masses of the carts. Record the masses and the starting position in Table 11.1.

REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA 5. Physics for Scientists and Engineers (with Modern Physics) 5th Edition by Raymond A. Serway & Robert J. Beichner @2000 Saunders College Publishing, Philadelphia

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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ Table 11.1: Results Additional Mass on Cart 1

Additional Mass on Cart 2

0

0

500 g

0

1000 g

0

500 g

250 g

m1

m2

Starting Position

x1

x2

x1 x2

m2 m1

Data Analysis 1.

For each of the cases, calculate the distances traveled from the starting position to the end of the track. Record the result in Table 11.1.

2.

Calculate the ratio of the distances traveled and record in the table.

3.

Calculate the ratio of the masses and record in the table.

Questions 1.

Does the ratio of the distances equal the ratio of the masses in each of the cases? In other words, is momentum conserved?

2.

When carts of unequal masses push away from each other, which cart has more momentum?

3.

When the carts of unequal masses push away from each other, which cart has more kinetic energy?

4.

Is the starting position dependent on which cart has the plunger? Why?

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Experiment 11.2: Conservation of Momentum in Collisions Purpose The purpose of this experiment is to qualitatively explore conservation of momentum for elastic and inelastic collisions.

Theory When two carts collide with each other, the total momentum of both carts is conserved regardless of the type of collision. An elastic collision is one in which the carts bounce off each other with no loss of kinetic energy. In this experiment, magnetic bumpers are used to minimize the energy losses due to friction during the collision. In reality, this “elastic” collision is slightly inelastic. A completely inelastic collision is one in which the carts hit and stick to each other. In this experiment, this is accomplished with the hook-and-loop bumpers on the carts. MATERIALS NEEDED: Track with Feet

Plunger Cart

Other Required Equipment

Collision Cart

Paper (for drawing diagrams)

Part I: Elastic Collisions

Figure 11.2 1.

Install the feet on the track and level it.

2.

Orient the two carts on the track so their magnetic bumpers are toward each other.

3.

Test cases A1 through A3 and B1 through B3 described below. Draw two diagrams (one for before the collision and one for after the collision) for each case. In every diagram, show a velocity vector for each cart with a length that approximately represents the relative speed of the cart.

A. Carts with Equal Mass Case A1: Place one cart at rest in the middle of the track. Give the other cart an initial velocity toward the cart at rest. Case A2: Start the carts with one at each end of the track. Give each cart approximately the same velocity toward each other. Case A3: Start both carts at one end of the track. Give the first cart a slow velocity and the second cart a faster velocity so that the second cart catches the first cart.

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B. Carts with Unequal Mass Put two mass bars in one of the carts so that the mass of one cart is approximately three times the mass (3M) of the other cart (1M). Case B1: Place the 3M cart at rest in the middle of the track. Give the other cart an initial velocity toward the cart at rest. Case B2: Place the 1M cart at rest in the middle of the track. Give the 3M cart an initial velocity toward the cart at rest. Case B3: Start the carts with one at each end of the track. Give each cart approximately the same velocity toward each other. Case B4: Start both carts at one end of the track. Give the first cart a slow velocity and the second cart a faster velocity so that the second cart catches the first cart. Do this for both cases: with the 1M cart first and then for the 3M cart first.

Part II: Completely Inelastic Collisions 1.

Orient the two carts so their hook-and-loop bumpers are toward each other. Push the plunger in all the way so it will not interfere with the collision.

2. Repeat test cases A1 through A3 and B1 through B3 and draw diagrams for each case. Questions 1.

When two carts having the same mass and the same speed collide and stick together, they stop. Is momentum conserved?

2. When two carts having the same mass and the same speed collide and bounce off of each other elastically, what is the final total momentum of the carts?

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125

EXPERIMENT NO.

Elective Activity

TORQUE & ROTATION OBJECTIVES: To determine the condition that must be satisfied for a body to be in rotational equilibrium. To find out the factors affecting torque. THEORY: Consider the plank shown in Figure 1. The fulcrum is exactly at the center. The blocks on the plank have the same weight (F1 = F2) and are placed at the same distance r (r1 = r2) away from the fulcrum. The length of the perpendicular drawn form the fulcrum (or pivot or axis of rotation) to the line of the force is called the lever arm. (This is also often called the moment arm). The cross product of the moment arm (r) and the force (F) is defined as torque (Γ). In equation form, torque is defined as: Γ=rxF

(Equation 1)

⇒

Γ = rF sin θ

Figure 1. A plank in equilibrium For simplicity, the direction of torque in this experiment will just be limited to two directions. A torque that results to a counterclockwise rotation will be assigned a negative (-) sign and for clockwise rotation, a positive (+) sign. Applying the definition of torque to the three forces present in Figure 1. yields the following: Force (F)

Moment arm (r)

sin θ

Torque (Γ)

F1

r1

sin 90o

-r1F1

F2

r2

sin 90o

+r2F2

F3

0

0

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For an object to be in complete equilibrium, it must satisfy two conditions. The first condition of equilibrium is that the net force acting on the object must be equal to zero. The second condition of equilibrium (also called rotational equilibrium) is that the net torque about an arbitrary axis of rotation must be equal to zero. The plank in Figure 1 is in complete equilibrium. It is in a condition of static equilibrium. Since it remains at rest, the net force acting on it is equal to zero. (Equation 2)

∑Fy = F3 – F1 – F2 = 0

The plank is also in a condition of rotational equilibrium. Since r1 = r2 and F1 = F2, the clockwise torque (r2F2) is equal to the counterclockwise torque (r2F2). (Equation 3) (Equation 4)

∑Γ = ∑Γclockwise – ∑Γcounterlockwise = r2F2 – r1F1 = 0 ∑Γclockwise = ∑Γcounterlockwise

Figure 2 Two situations where the plank system may not be in equilibrium Moving the left block near the fulcrum as in Figure 2(a) would tend to rotate (or tilt) the plank in a clockwise manner even though the forces acting on the plank remain the same. The clockwise torque is not equal to the counterclockwise torque since the moment arm on the left is now less than the moment arm on the right side. Adding another block on the right side of the fulcrum as in Figure 2(b) would also tend to rotated (or tilt) the plank in a clockwise manner even through the moment arm on the left is equal to the moment arm on the right. The clockwise torque is not equal to the counterclockwise torque since the left force (weight) is now less than the force (weight) on the right side. MATERIALS: Weights, weight holder, clamp, iron stand, meter stick, weighing scale

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Figure 3. The meter stick with the other accessories

REFERENCES 1. Physics Laboratory Experiments 4th Edition, Jerry D. Wilson @ 1994 D.C. Heath and Company; Lexington, Massachusetts 2. Laboratory Manual In Conceptual Physics 2nd Edition, Bill W. Tillery @ 1995 Wm. C. Brown Communications, Inc. Dubuque, IA 3. Laboratory Experiments in College Physics 7th Edition, Cicero H. Bernard, Chirold D. Epp @ 1995, John Wiley and Sons, Inc. New York 4. Experiments in Physics 2nd Edition, Peter J. Nolan, and Raymond E. Bigliani @ 1995 Wm. C. Brown Publishers; Dubuque, IA PROCEDURE: A. Center of Gravity Locate the center of gravity of the meter stick by placing it in the clamp as shown in Figure 3. Adjust the meter stick (without weights) until it is in equilibrium. B. Rotational Equilibrium of Balanced Forces On the balanced meter stick, place an equal amount of loads (about 200 g) on both sides of the meter stick at a distance of about 25 cm from the fulcrum. (Include the weight of the clamp and weight holder in all of the forces in this exercise.) Adjust slightly the position of the loads to maintain equilibrium. Record the forces and their moment arms. Calculate and record the clockwise torque and the counterclockwise torque. How do they compare? Record the percentage difference between them. Repeat the above procedure for two more trials, adding 50 g each time to the original loads.

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C. Rotational Equilibrium of Unbalanced Forces On the balanced meter stick, place two unequal amounts of loads (about 100 g and 250 g) on both sides of the meter stick at a distance of about 25 cm from the fulcrum. (Include the weight of the clam and weight holder in all of the forces in this exercise.) Adjust the position of the loads to maintain equilibrium. Record the forces and their moment arms. Calculate and record the clockwise torque and the counterclockwise torque. How do they compare? Record the percentage difference between them. Repeat the above procedure for two more trials, adding 50 g each time to the lesser load.

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__________________________________________________ __________________________________________________ __________________________________________________ __________________________________________________

__ ___________________________________ __ ___________________________________ DATA SHEET: A. Center of Gravity of the meter stick system: ________cm mark

B. Rotational Equilibrium of Balanced Forces Trial #

Mass Force (kg) (N)

Moment Arm (m)

Torque (N.m) Clockwise

Counter Clockwise

Percentage Error

First Second Third

C. Rotational Equilibrium of Unbalanced Forces Trial #

Mass Force (kg) (N)

Moment Arm (m)

Torque (N.m) Clockwise

First Second Third

131

Counter Clockwise

Percentage Error

Physics Laboratory Manual 1

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