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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

Extracts from this document...

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

...read more.

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.

...read more.

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

  1. A heavier mass could be used to obtain a smoother motion.
  2. If possible, more trials can be done to average out the random errors and obtain a better result.

...read more.

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