- Level: AS and A Level
- Subject: Science
- Word count: 2950
Physic lab report - study the simple harmonic motion (SHM) of a simple pendulum and to investigate the phase relationship between the displacement, velocity and acceleration, and to investigate how acceleration is related to displacement in a simple harmo
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Introduction
Ivan Liu Chun Pok
6A(11)
Group 8
Laboratory report: Studying a simple harmonic oscillation
Objectives:
To study the simple harmonic motion (SHM) of a simple pendulum and to investigate the phase relationship between the displacement, velocity and acceleration, and to investigate how acceleration is related to displacement in a simple harmonic motion.
Apparatus:
- half metre rule
- a light string
- pendulum bob
- video camera with tripod stand
- computer with Motion Video Analysis (MVA) software and Microsoft Excel installed
Experimental design:
Fig. 0
Theory:
For an object or mass moving in a simple harmonic motion, the displacement, velocity and acceleration change periodically in both magnitude and direction. The acceleration in particular is always proportional to its displacement from the equilibrium position and must always be directed towards the equilibrium point. Mathematically it can be expressed as
a = -kx, where k is a constant and x is the displacement from the equilibrium point.
Also for a simple harmonic oscillation, the period or frequency of oscillation is independent of the amplitude of the motion.
In Figure 1, x is the displacement of the pendulum bob from the equilibrium point Q. Points P and R are points where the maximum displacement (amplitude A) can be obtained. Theoretically, the following equations are true for S.H.M.:
When the motion starts at the equilibrium position (point Q)
x = A sin ωt where ω is angular velocity
v = ω A cos ωt
a = - ω2A sin ωt
Period T = 2 π / ω
Fig. 1
When the motion starts at the position
Middle
8.15E-03
-2.04E-03
-4.59E-01
-1.83E+00
<BR />2.30E+01
1.47E+00
-1.63E-02
0.00E+00
-3.67E-01
1.38E+00
<BR />2.40E+01
1.53E+00
-4.69E-02
2.04E-03
-4.59E-01
-1.38E+00
<BR />2.50E+01
1.60E+00
-8.36E-02
8.15E-03
-5.50E-01
-1.38E+00
<BR />2.60E+01
1.67E+00
-1.08E-01
1.43E-02
-3.67E-01
2.75E+00
<BR />2.70E+01
1.73E+00
-1.32E-01
1.63E-02
-3.67E-01
-7.58E-06
<BR />2.80E+01
1.80E+00
-1.59E-01
1.63E-02
-3.97E-01
-4.59E-01
<BR />2.90E+01
1.87E+00
-1.81E-01
1.83E-02
-3.36E-01
9.17E-01
<BR />3.00E+01
1.93E+00
-2.02E-01
2.04E-02
-3.06E-01
4.59E-01
<BR />3.10E+01
2.00E+00
-2.26E-01
2.65E-02
-3.67E-01
-9.17E-01
<BR />3.20E+01
2.07E+00
-2.41E-01
2.85E-02
-2.14E-01
2.29E+00
<BR />3.30E+01
2.13E+00
-2.47E-01
2.85E-02
-9.17E-02
1.83E+00
<BR />3.40E+01
2.20E+00
-2.59E-01
3.06E-02
-1.83E-01
-1.38E+00
<BR />3.50E+01
2.27E+00
-2.65E-01
3.06E-02
-9.17E-02
1.38E+00
<BR />3.60E+01
2.33E+00
-2.71E-01
3.06E-02
-9.17E-02
1.86E-07
<BR />3.70E+01
2.40E+00
-2.55E-01
3.06E-02
2.45E-01
5.04E+00
<BR />3.80E+01
2.47E+00
-2.43E-01
2.85E-02
1.83E-01
-9.17E-01
<BR />3.90E+01
2.53E+00
-2.32E-01
2.85E-02
1.53E-01
-4.59E-01
<BR />4.00E+01
2.60E+00
-2.20E-01
2.65E-02
1.83E-01
4.59E-01
<BR />4.10E+01
2.67E+00
-1.98E-01
2.45E-02
3.36E-01
2.29E+00
<BR />4.20E+01
2.73E+00
-1.79E-01
2.24E-02
2.75E-01
-9.17E-01
<BR />4.30E+01
2.80E+00
-1.57E-01
2.24E-02
3.36E-01
9.17E-01
<BR />4.40E+01
2.
Conclusion
As the two ends of the half-metre rule may not be marked accurately in the MVA software, the distance marked may not be exactly 0.5 m. Same as error (2), as the MVA software requires the setting of the end points of the half-metre rule as a reference to locate the displacement, the displacement at each time interval does not reflect the true value of the displacement.The position of mass marked for each time interval may not be the same for all time intervals
It is difficult to locate the mass at the same position for each time interval, therefore the displacement obtained is not accurate for each time interval.
There may be a damping effect by air resistance
Air resistance exists, hence a damping force acts on the mass in motion, resulting in smaller and smaller amplitude obtained and also causing deviations in displacement.
The spring may not be perfectly elastic
As the spring provided may not be perfect, the whole motion may not be entirely a simple harmonic motion. The graphs obtained from the experimental results may not truly reflect the characteristics of a simple harmonic motion.
Conclusion
The velocity leads the displacement by a quarter of the cycle, and the acceleration leads the velocity also by a quarter of the cycle.
Also, the acceleration is directly proportional to displacement in a simple harmonic motion and is in an opposite direction to x.
Possible improvements of the experiment
- A heavier mass could be used to obtain a smoother motion.
- If possible, more trials can be done to average out the random errors and obtain a better result.
This student written piece of work is one of many that can be found in our AS and A Level Mechanics & Radioactivity section.
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