1.0 ABSTRACT For experiment 3A, the experiment is conduct to investigate investigat e the relationship between the spring constant and the mass on a spring. The proportionality constant k, is called spring constant or spring stiffness constant.This experiment is about to find the period of oscillations of spring by using different mass on a spring. If the value of k increase, the force needed to stretch the spring in a given distance increase. The step to conduct this experiment is when the mass had been steady, released the rod smoothly, so that the mass and the rod will oscillates up and down. For experiment 3B, this experiment is conduct to investigate the behavior of the simple pendulum.The displacement of the
pendulum along the arc is given by X = Lθ, where θ is the angle the cord makes with the vertical and L is the length of the cord. The step to conduct this experiment is set the mass to swing and keep the angle of the swing reasonably small. The time taken for the oscillations was taken. If the rest oring force is proportional to X or to θ, the motion will be simple harmonic. The restoring force is the net force on the bob, equal to the component of the weight, mg, tangent to the arc.
2.0 INTRODUCTION A simple pendulum consists of a small object (the pendulum bob) suspended from the end of a lightweight cord. The motion of a simple pendulum swinging back and forth with neglible friction resembles simple harmonic motion. The pendulum bob oscillated along the arc of a circle with equal amplitude on either side of its equilibrium point, and as it passes through the equilibrium point (where it would hang vertically) it has the maximum speed. The restoring force is proportional to its displacement and it is undergoing Simple Harmonic Motion. The concern of this experiment is to determine the period of the motion (T).
In the first experiment, Hooke’s Law is used. By using this law, force exerted by a spring is proportional to the distance beyond its normal length to which it is stretched. For the second experiment, the displacement of the pendulum along of the arc is given by x = Lθ, where θ is the angle of the cord makes with the vertical and L is the length of the cord. If the restoring force is proportional to x or to θ, the motion will be simple harmonic motion. The restoring force is the net force on the bob, equal to the component of the weight, mg, tangent to the arc.
3.0 OBJECTIVE The purpose of this experiment is to become familiar with the resulting motion, this is called Simple Harmonic Motion by applying the simple pendulum behavior, and to study the relationship between the mass on a spring and the spring constant.
.
4.0 THEORY Experiment 3A : Basiccaly, Simple Harmonic Motion is happen when the object vibrates or oscillates back and forth. It is affected by gravitional force .It the experiment about mass on a spring, the mass is affected by gravity. By this statement, it can be shown that :F = mg F = Force m = mass g = grativional acceleration
It such that, this experiment clearly describes about Hooke’s Law which state that the force exerted by a spring is proportional to the distance beyond its normal length to which it is stretched. This idea could be write mathematically as :F = -kx F = Force exerted on spring -k = spring constant x = displacement of the end of the spring from its equilibrium position From the equation, we can measure the spring constant. -k = F/x
Where the negative sign indicates that restoring force is always in the direction opposite to the displacement, x.
When the simple harmonic motion of a mass (M) on a spring is analyzed
mathematically using Newton’s 2 nd law, the period of the motion (T) is found to be :
√
T=2
Experiment 3B Simple harmonic motion plays an important role in our daily life. It can be found that from vibration and oscillation, simple harmonic motion can be occurred by the presence of vibration and oscillation. A swinging pendulum is one of the example that can be seen in our daily life. It affected by a mass that when mass increase, period of oscillation also increase. T is directly proportional to m. From figure 3.2, it can be seen that the mass is affected by gravity called gravitional force. Using the two congruent triangles in the diagram, it can be seen that F x = mgsinθ, and that the displacement of the mass from its equilibrium positions is an arc whose distance, x, is approximately L tanθ . If the angle θ is reasonably small, then it is very nearly true that sin θ
= tanθ. Therefore, for a small swings of the pendulum, it is
approximately true that F x = mgtanθ = mgx/L. Since Fx is a restoring force, the equation could be stated more accurately as F x = -mgx/L. Comparing this equation with the equation for a mass on a spring (F= -kx), it can be seen that the quantity mg/L plays the same mathematical role as the spring constant. On the basis similarity, the period of motion for a pendulum is just:
= 2
T=2
Where m is the mass, g is the acceleration due to gravity, and L is distance from the pivot point to the center of mass of the hanging mass.
5.0 APPARATUS Experiment A :
Experiment board
Mass hanger
Stopwatch
Spring balance
Masses
Experiment B :
Experiment board
Mass hanger
String
Pivot
Masses
6.0 PROCEDURES Experiment A: Mass on a spring 1. The spring constant, k for the spring was measured in the spring balance. K=___________ (Newtons/meter) 2. The apparatus were set up as shown in figure 10.1, with 120 grams on the Mass Hanger (125 grams total mass, including the hanger). 3. Been sure that the spring balance was vertical so that the rod hangs straight down through the hole in the bottom of the balance. This is important to minimize friction against the side as the mass oscillates. 4. The rod was pulled down for few centimeters. The mass was steady and the rod was letting go. Practice until the rod can released smoothly, so the mass and the rod can oscillate up and down and there is no rubbing of the rod against the side of the hole. 5. The mass oscillating was set up. The time taken was measured for at least 10 full oscillating. (The time was measured for as many oscillations can be conveniently counted before the amplitude of the oscillation becomes too small). 6. The mass, time, and the number of oscillations counted were recorded. The total time was divided by the number of oscillations observed to determine the period of oscillations. (The period is the time required for 1 complete oscillation). This value was recorded in the table. 7. The measurement was repeated 3 times. The period for each measurement were calculated. Then, three period measurements was added together and divided by 3 to determine the average period over all three measurements. 8. The equation given at the beginning of experiments was used to calculate a theoretical value for the period using each mass value. (Mass value used in the
equation must be in kg since the spring constant is in units of newtons/m). This value was recorded in the table below in the results. Experiment B: The Pendulum 1. A Mass Hanger was hanging from the pivot as shown in the figure 11.1. 2. The mass swinging was set, but the angle of the swing reasonably was kept small. 3. The time takes was measured for at least 30 full oscillations to occur. 4. The mass, distance L, the time and the number of oscillations counted were recorded in the Table 11.1. 5. The total time was divided by the number of oscillations observed to determine the period of the oscillations. (The period is the time required for one complete oscillation). This value was recorded in the table. 6. The measurements were repeated 5 times. The periods were calculated for each measurement. Then five period measurements were added together and divided by 5 to determine the average period over all three measurements. 7. The experiment was repeated using different mass. 8. The equation given at the beginning were used to calculate theoretical value for the period in each case (g=9.8 N/m; been sure to express L in meters when plug into the equation). This value was recorded in the table below in the results.
7.0 RESULTS AND DISCUSSION 7.1 RESULTS 7.1.1 Experiment 3A: Mass on spring Spring constant for experiment A:
=
= 36.03
⁄
Spring constant for experiment B:
= = 35.73 ⁄ Spring constant for experiment C:
= 36.15
⁄
Table 3.1 Periods against Mass Experiment
Mass,
Number of
M/kg
oscillations
Time, t/s
Average
Period, T/s
Period,
(calculated)
̅/s A
0.125
10
3.30
0.356
0.370
0.408
0.440
0.487
0.496
3.11 3.24 4.02 4.15 B
0.175
10
4.20 4.47 3.84 3.77 4.14
C
0.225
10
4.92 5.01 4.96 4.91 4.53
Percentage error for experiment A =
= 3.88%
Percentage error for experiment B =
= 7.27%
Percentage error for experiment C =
= 1.81%
7.1.2 Experiment 3B: The pendulum Table 3.2 Period against mass Experiment
Mass,
Distance,
Number of
Time,
Average
Period, T/s
M/kg
L/m
oscillations
t/s
period, /
̅
(calculated)
s A
0.05
0.15
30
23.31
0.762
0.777
0.744
0.777
20.88 23.17 22.63 24.27 B
0.10
0.15
30
21.10 22.31 22.96 23.04 23.17
Table 3.3 Period against distance of string Experiment
Mass,
Distance,
Number of
M/kg
L/m
oscillations
Time, t/s
Average
Period, T/s
̅
(calculated)
period, / s
A
0.05
0.15
30
23.31
0.762
0.777
1.069
1.099
20.88 23.17 22.63 24.27 B
0.05
0.3
30
31.78 32.63 32.15 32.24 31.55
Percentage error from Table 3.2A =
= 1.93%
Percentage error from Table 3.2B =
= 4.25%
Percentage error from Table 3.3A =
= 1.93%
Percentage error from Table 3.3B =
= 2.73%
7.2 DISCUSSION 7.2.1 Experiment 3A: Mass on spring Whenever an object is acted on by a restoring force that is proportional to the displacement of the object from its equilibrium position, the resulting motion is called Simple Harmonic Motion. It is assumed that the magnitude of restoring force, F is directly proportional to the displacement, x the string has been stretched or compressed
from the equilibrium position, as stated by Hooke’s law:
(Equation 3.1)
The theoretical value for the period was calculated using the equation below which is derived from the Equation 3.1:
(Equation 3.2)
The period depends on the mass, m and the spring stiffness constant, k. From Equation 3.2, it is clear that the larger the mass, the longer the period; and the stiffer the spring (larger k), the shorter the period. This makes sense as a larger mass produces more inertia and hence slower response, resulting in smaller acceleration. If the k is larger, it means that the force is greater and hence the response is quicker, resulting in larger acceleration. The Equation 3.2 is not a direct proportion because the period varies as the square root of m/k. From the Experiment 3A, mass was varied to show the change in period which represents the motion of the spring. Based on the results tabulated in Table 3.1, in experiment A, the theoretical period, T = 0.370 s while the actual period which is average period obtained from the experiment,
̅ = 0.356 s. The percentage
error for this experiment is 3.88%. The theoretical value is does nearly accurately predict the experimental value. This must be because of the errors occurred during the
experiment such as parallax error, air resistance and frictional force exerted by the mass of the pendulum on the board. 7.2.2 Experiment 3B: The pendulum A simple pendulum of length, L approximates simple harmonic motion if its amplitude is small and friction can be ignored. For small amplitudes, its period is then given by
(Equation 3.3)
. The mass, m of the
Where g is the acceleration of gravity which is equal to 9.8 m/
pendulum mass does not appear in the equation. Thus, the period of a simple pendulum do not depend on the mass of the pendulum mass. This was proved by the results obtained from Table 3.2, as the theoretical period, T = 0.777 s when both mass of 0.05 kg and 0.10 kg were used. Moreover, it does not depend on the amplitude as long as the amplitude
is small. Because a pendulum does not undergo accurately
simple harmonic motion, the period does depend slightly on the amplitude--the more so for large amplitudes. The accuracy of the pendulum would be affected, after many swings, by the decrease in amplitude due to friction. During Experiment 3B, the
amplitude used, = 0. Hence, the experimental values obtained were slightly differing from the theoretical values. This was shown by the results tabulated in Table 3.3 where the theoretical period, T = 0.777 s where as the experimental period which is average period,
̅ = 0.762 s resulting the percentage error is equal to 1.93%. Based on Table
3.3, where period was affected by distance, L, it is assumed that when L increases, period (theoretical period) increases.
8.0 SAMPLE OF CALCULATIONS 8.1 Experiment 3A: Mass on spring Spring constant for experiment A:
= = 36.03 Period,
⁄
for experiment A:
= = = 0.36 s Average period for experiment A:
̅ = = = 0.356 s Period, T calculated from Equation 3.2 for experiment A:
= = 0.370 s Percentage error for experiment A =
= 3.88%
8.2 Experiment 3B: The pendulum Period,
for experiment 3.2A:
= = = 0.777 s Average period for experiment 3.2A:
̅ = = = 0.762 s Period, T calculated from Equation 3.3 for experiment 3.2A:
= = 0.777 s Percentage error from Table 3.2A =
= 1.93%
9.0 CONCLUSION In experiment 3A, when mass of the pendulum changed, the value of the period of the oscillations changed. The higher the mass of the pendulum, the higher the period of the oscillations. In experiment 3B, the period of oscillations depends on the length of the string. The shorter the length of the string, the shorter the period of the pendulum. Errors that occur during these experiments were air resistance, inaccuracy of reading and frictional force exerted on the board by the mass of the pendulum. Theoretical value for the period is slightly accurate to the experimental value. So, the equation given is valid. Percentage error in experiment 3A was 3.88% while in experiment 3B was 1.93%. Simple Harmonic Motion is the resulting motion that used in this experiment. Applications of these experiments are rubber bands and bungee cords.
10.0 RECOMMENDATION AND REFERENCES Recommendation: 1) Make sure the stopwatch work efficiently. 2) Switch off the fan. 3) Make sure the mass of pendulum is not close to the wall of the board. 4) Express L in meters when the equation is used. References: 1) D. C. Giancoli (2005), Physics, Sixth Edition, Pearson Education, Inc, NJ, p 287. 2) T. B. Cole (2004), Physics, Sixth Edition, David Harris, USA, p 463. 3) R. A. Serway (1996), Physics, Fourth Edition, John Vondeling, New York, p 361.
