Physics Lab Report
Determining the effects of amplitude, length and, mass on the
period of simple pendulum
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Simple Pendulum
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Introduction
The aim of this experiment was to determine the effect of length on the period of oscillation of a simple pendulum and to find the acceleration due to gravity. The word pendulum came from the Latin word which means hanging. A pendulum is a hanging
object from a fixed point which it swings freely when it pulled back and released. Earth gravity pulls
the hanging object toward the centre of the Earth and because the inertia it keeps moving or if it at
rest it remain at rest.
In this experiment the length of the pendulum is changed, and the time period is measured. Using
this data, a graph can be plotted in order to find the acceleration due to gravity.
Theory
When a simple pendulum is displaced from its equilibrium position, there will be a restoring force that moves the pendulum back towards its equilibrium position. As the motion of the pendulum carries it past the equilibrium position, the restoring force changes its direction so that it is still directed towards the equilibrium position. If the restoring force F is opposite and directly proportional to the displacement x from the equilibrium position, so that it satisfies the relationship
F = - k x (1) Where K is a constant. Then the motion of the pendulum will be simple harmonic motion and its period can be calculated using the equation for the period of simple harmonic motion, T. m is the mass.
T = 2π √ 𝑚
𝑘 (2)
It can be shown that if the amplitude of the motion is kept small, Equation (2) will be satisfied and the motion of a simple pendulum will be simple harmonic motion, and
The restoring force for a simple pendulum is supplied by the vector sum of the gravitational force on the mass. mg and the tension in the string, T. The magnitude of the restoring force depends on the gravitational force and the displacement of the mass from the equilibrium position. Consider Figure 1 where a mass m is suspended by a string of length l and is displaced from its equilibrium position by an angle θ and a distance x along the arc through which the mass moves. The gravitational force can be resolved into two components, one along the radial direction, away from the point of suspension, and one along the arc in the direction that the mass moves. The component of the gravitational force along the arc provides the restoring force F and is given by:
F = - mg sinθ (3)
Figure 1
l
m
θ
Simple Pendulum
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Where g is the acceleration of gravity, θ is the angle the pendulum is displaced, and the minus sign indicates that the force is opposite to the displacement. For small amplitudes where θ is small, sinθ can be approximated by θ measured in radians so that Equation (3) can be written as
F = - mg θ (4) The angle θ in radians is xl, the arc length divided by the length of the pendulum or the radius of the circle in which the mass moves. The restoring force is then given by
𝐹 = − 𝑚𝑔 𝑥
𝑙 (5)
and is directly proportional to the displacement x and is in the form of Equation (1) where k = mg/ l. Substituting this value of k into Equation (2), the period of a simple pendulum can be found
T = 2π √ 𝑙
𝑔 (6)
Therefore, for small amplitudes the period of a simple pendulum depends only on its length and the value of the acceleration due to gravity. (Serway, 2010) As we can see from equation six that the amplitude and the mass did not effects the period.
However, the length of the string effect the period of oscillation and we can have straight line graph
for 𝑡2 𝑣𝑠 𝑙
Health and safety
This experiment is not dangerous but it can be if you are not careful. Tide the string on the hanging
mass so that does not fall on your feet and always keep your feet away from the mass so if it fall it
did not harmful your feet.
Apparatus
• Retort stand with boss and clamp
• String, card board strips
• Slotted 100g masses
• Mass hanger
• Protector
• timer
• Scale
• Metre rule
Simple Pendulum
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Method
Hang the string on the top of the retort stand and attach the mass on the other end of the string.
Measure the length of the string from the top of the retort stand to the centre of mass of the
hanging mass. Left the hanging mass to angle as shown in the diagram. Leave the mass and start the
timer at the same time. After the tenth period end, stop the stop the timer. Keep the angle and the
hanging mass fixed and repeat the procedure several times for different lengths.
Results
Mass: 0.0978 ± 0.0001 kg
Amplitude: 20 ± 1 θ
Length /m
±0.001
Time for Ten Oscillations /s
± 0.1
Average /s
Time for one Oscillation /s
Time Period Squared /s2
0.974 19.8 19.9 19.8 19.9 1.99 3.96
0.865 18.8 18.7 18.9 18.8 1.88 3.53
0.798 17.6 17.7 17.5 17.6 1.76 3.10
0.663 16.2 16.2 16.3 16.3 1.63 2.66
0.564 14.9 15.0 15.1 15.0 1.50 2.25
0.406 12.5 12.7 12.6 12.6 1.26 1.59
Retort stand
angle
string
mass
Protector
Simple Pendulum
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Analysis
Graph 1:
Calculations: From graph 1: Gradient of the line equal to 4π2 /g Using the trendline equation the gradient is 4.15 to 3 S.F. Hence 4.15 = 4π2 /g so g = 4π2 / 4.15 = 9.51 The acceleration due to gravity from the experiment is: 9.51 m s-2 The trendline is a good fit to the points, only one point is slightly below the trend line. This means there is very little experimental error, and it is not worth finding the minimum and maximum gradients for the trendline. There is an uncertainty in the result and this can be calculated as follows: g = l 4π2 T2
Uncertainty in length is ± 0.001m which gives an average percentage uncertainty of 0.001 x 100 = 0.15 % (0.974 +0.406)/2 Uncertainty in Time period is ± 0.1 s Which gives an average percentage uncertainty of 0.1 x 100 = 6.2 % (1.99 + 1.26)/2 For time period squared percentage uncertainty is 6.2 + 6.2 = 12.4 % Average uncertainty in g = 0.15 % + 12.4 % = 12.55 % or 12.6% to 3 S.F.
y = 4.1465x - 0.1032
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0 0.2 0.4 0.6 0.8 1 1.2
T im
e P
e ri
o d
S q
u a
re d
/ s2
Length /m
Time Period Squared versus Length
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12.6 % of 9.51 is 1.2 So the experimental value of g = 9.51 ± 1.2 m s-2
Errors Errors form the measuring equipment in the time of one period. We used the scale and the error
from the scale is ±0.000005.the ruler error is ±0.0005. ±0.5 is the error from the protector. Final
error is ±0.500505. The error in the g is ±0.501005 because we squared the time and the length.
Discussion Human reaction causes the major error in this experiment by measurement of the string and the
time of the swing was measured by stopwatch. The point of pivot creates a lot of friction and the
stand was shaking which caused errors. We can avoid the error from the human reaction by
measuring the time using the beam timer to measure the periods instead of the timer. Repeating the
experiment 5 times will give us better results than 3 times.To reduce the error from the length
measurements we should take accurate measurements of the center of the hanging mass and by
attaching the stand to big mass it will not shake.
Conclusion The aim of the experiment was to determine effect of amplitude, length of string and, mass
hanging on the period of oscillation of simple pendulum. The results were reliable because it give us
good approximation to the gravity constant g. The length of the string effect the period by plotting
the graph of t vs l It did not give us a straight line graph but by plotting t^2 vs l we got the straight
line graph and the amplitude and the hanging mass do not have any effect on the period as we
expected . So the hypothesis was completely correct.
Simple Pendulum
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References
Parsons, B., 2009. [Online]
Available at: http://www.bukisa.com/articles/34123_the-history-of-the-pendulum#ixzz2lZnhVv9u
[Accessed 23 novermber 2013].
Serway, R. A., 2010. physics for scientist and engineers. 8th ed. belmont: Lachina Publishing Services.
National Institute of Standards and Technology. 2009. The NIST Reference on Constants, Units and Uncertainty [Online] Available at: http://physics.nist.gov/cgi-bin/cuu/Value?gn [Accessed 23 November 2013]