COLLEGE OF SCIENCES Department of Applied Physics
LABORATORY EXPERIMENTS IN PHYSICS (1)
Vectors
Prepared by :
BASSAM RASHED
36
Vectors Objectives: In this experiment you should learn the definition of a vector, and how to represent it in space. Also, you should learn how to apply the rules for vector addition both graphically and analytically.
Apparatus: Force table with three to four pulleys, mass hangers, slotted masses, string, protractor, ruler, and sheets of graphic papers .
37
Theory : Physical quantities can be classified into two main categories ; scalar and vector quantities . A scalar quantity requires only a magnitude to be given , whereas a vector quantity requires both magnitude and direction for complete description . The direction of a vector quantity is as important as the magnitude and without it the quantity is meaningless. Vector may be represented graphically as a directed line segment . The length of the line represents the vector’s magnitude and the line’s angle with respect to some coordinate system specifies the vector’s direction. As an example of the process of vector addition consider the case of several forces with different magnitudes and directions which act at the same point . It is desired to find the net effect produced by the several forces by finding a single force which is equivalent in its effect to the effect produced by the several applied forces. That single vector is called the resultant vector of the several applied vectors . This resultant theoretically by a special addition process known as vector addition. The resultant vector ( R ) of two vectors as an example A & B can be found by two methods analytical and graphical method . In the analytical method each vector such as ( A ) which makes an angle ( ) with horizontal x- axis is first resolved into two components . Those components are horizontal or x- component (Ax) and vertical or y- component (Ay ). Those components are given by : A A Ax = A cos ( ) , Ay = A sin ( ) Ax Those components are at right angles , and thus the magnitude of the resultant can be found from Pythagorean theorem. Consider the case of three vectors A , B , C , firstly we resolve each vector into two components , then taking the algebraic y
38
sum of each of the components of the three vectors leads to the following : R x = Ax + Bx + Cx & R y = Ay + B y + Cy The magnitude of the resultant vector , R , is found to be the following because the components R x and R y are at right 2
2
R ( R x ) ( R y ) and the angle ( ) angles: resultant makes with x- axis is given by the following :
tan
-1
(
R y R x
)
that the
.
In the graphical addition process known as the polygon method one of the vectors is first drawn to scale . Then each successive vector to be added is drawn with its tail starting at the head of the preceding vector . The resultant vector is then the vector drawn from the tail of the first arrow to the head of the last arrow. The polygon method is illustrated for the case of three vectors as follows: C
B
A Firstly we draw the vector A with the same angle that it makes with the positive x – axis choosing the appropriate and the same scale for all vectors . Then B is drawn at the proper angle ( ) relative to A, and C is then drawn at the proper angle ( ) relative to B. Finally the resultant R is the vector connecting the tail of vector A and the head of vector C As shown in the figure below :
39
C R
B
A
So, you measure the length of the vector R which represents its magnitude and measure the angle ( ) that the vector makes with the positive x- axis. The process of experimentally determining the value of the resultant of several forces is complicated by the fact that when nonzero resultant force acts on an object it tends to be accelerated . Thus another force must be applied to produce an equilibrium . for example , if two known forces , F 1 and F2 , are applied to some object , they will have some resultant F R . In order to keep the object in equilibrium , a force equal in magnitude and opposite in direction of FR must be applied . The force applied in order to produce equilibrium is called the equilibrant force F E . so , the F E is the equilibrant force that must be applied in order to keep an object in equilibrium . The magnitude and direction of this F E can be found by trial and error experimentally . The resultant force F R can be found from knowledge that FR and FE have the same magnitude but opposite directions. The force table used in the experiment is designed to allow application of forces of any chosen magnitude at any chosen angle . The force is provided by the gravitational attraction on masses that are attached to a ring by string passing over a pulley. each force is applied over a separate pulley , and pulley positions can be adjusted to any desired position around a circular plate. It is 40
important to note that the force applied when a mass is hung on the string is equal to the weight of the mass. The weight (in Newtons) is equal to the product of the mass (in kilograms) multiplied by the acceleration due to gravity (9.8 m / sec 2).
Procedure : A : Resul tant of two vector s 1
1) make sure that the top surface of the force table is horizontal , and the rod in the center of table is at the center of the ring when the two hangers are at zero and 1800 scale. 2) With the first hanger on 20 0 scale slot any mass on it then find F1. 3) Add a mass to the second hanger and find F 2 , then move it by means of its clamp , and fix it at 90 0 scale of the table. 4) Find the force that is added to the third hanger to balance the above two forces . This force is known as F E. 5) Find the resultant force FR ( magnitude and direction ) 6) Record your data for masses and F1 , F2, F E , FR , , in table 1 in your lab report.
A : Resul tant of th r ee vectors : 2
1) With the first pulley on scale 30 0 slot any mass on it then find F1. 2) Add a mass to the second hanger and find F 2 , then fix it at 1000 scale. 3) Add a mass to the third hanger and find F3 , then fix it at 1450 scale. 4) Following a procedure like that in the previous part , find F E and FR . 5) Record your data in table 2 in your lab report. 41
Data Analysis : 1 ) For part A1 , use analytical method to find F1x , F2x , F1y , F2y , FRx , FRy , F R , and that the resultant makes with x - axis .
2) Calculate the percentage error of the magnitude of the experimental value of FR compared to analytical solution for F R . 3) Use graphical (parallelogram) method to find the resultant of the two forces . 4) Calculate the percentage error of the magnitude of the graphical solution for FR compared to analytical solution for F R 5) For part A2 , use analytical method to find : F1x , F2x , F3x , F1y , F2y , F3y
, FRx , FRy ,
FR , and that the resultant makes with x - axis .
6) Calculate the percentage error of the magnitude of the experimental value of FR compared to analytical solution for FR . 7) Use graphical ( parallelogram ) method to find the resultant of the three forces . 8) Calculate the percentage error of the magnitude of the graphical solution for FR compared to analytical solution for F R .
42
COLLEGE OF SCIENCES Department of Applied Physics
LABORATORY EXPERIMENTS IN PHYSICS (1)
Vectors STUDENT LAB REPORT
Name: ……….……………….……….…………….…… . ID No.:…………………....…… Section:………..……... . Partner’s name:
………….……………….….. .
Date:…………..………… . Instructor’s name
………………………… .
43
Vectors Force F1 F2 Equilibrant FE Resultant FR
Analytical solution Force ( N ) Direction x-component 200 900
Force F1 F2 FR FR
Table 1 Mass ( Kg) Force ( Newtons )
Direction 200 900
y-component
Percentage error of FR (use the analytical result as real value).
Graphical method for the resultant of two force vectors
Percentage error of FR (use the analytical result as real value).
44
Force F1 F2 F3 Equilabrant FE Resultant FR
Analytical solution Force ( N ) Direction x-component 300 1000 1450
Force F1 F2 F3 FR FR
Table 2 Mass ( Kg ) Force ( Newtons )
Direction 300 1000 1450
y-component
Percentage error of FR (use the analytical result as real value).
Graphical method for the resultant of three vectors
Percentage error of FR (use the analytical result as real value).
45
Questions : 1) What is the difference between vector and scalar quantity ?
2) Classify each of the following physical quantities as vectors or scalars : a) Volume : b ) Force : c ) density : d ) velocity
e ) distance
g ) mass
h ) speed
f ) acceleration i) weight
3) What are the conditions of equilibrium for given forces ?
4) What are the conditions for the two vectors to be equal ?
5) Two forces , one of 2N and the other of magnitude 3N , are applied to the ring of a force table . The direction of both forces are unknown . Which best describes the limitations on , FR the magnitude of resultant a ) FR 5N d ) 1N
b ) 2N
FR
FR
5N
3N
c ) e ) FR
FR 3N
2N
6) If the ring touches the rod , is the system at equilibrium or not, why ?
46
7) Which of the following graphs represent a resultant force of zero value acting on the same point of the object : why ?
(a)
(b)
(c)
(d)
8) List the possible sources of error in the experimental determination of the resultant force using the force table .
9) Conclusion:
47
