Name: Yu Wai So (19) 6S

Experiment Report: Studying a simple harmonic oscillator

Objective

The simple harmonic motion of a pendulum can be studied by attaching a ticker-tape to a pendulum bob and analyzing the dots marked on the tape.

Theory

In this experiment, a string was used to suspend a 0.5 kg mass.

Refer to the diagram above,

Considering the tangential force on the mass,

∴The oscillation is simple harmonic.

Therefore, we can find out more on simple harmonic motion by analyzing the ticker-tape we obtained after the experiment.  

Apparatus

1. 0.5 kg ringed mass
2. 1.5 m length of string
3. Ticker-tape timer
4. Ticker-tape
5. Low voltage power supply (a.c.)
6. Retort stand and clamp

Procedure

1. The apparatus as shown in the figure was set up. A pendulum was suspend by a string and was attached to a retort stand and clamp. A ticker- tape was attached to the mass and was inserted to the ticker-tape timer. The mass was pulled to one side. The timer was switched on and the mass was allowed to swing to the other side.

2. After the pendulum reached its highest position, the ticker-tape timer was switched off and the ticker-tape was detached from the mass. The dots marked on the tape were examined. During the oscillation, the pendulum bob accelerated and then decelerated when it was approaching its highest position.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Discussion

        In the experiment, the string used is 1.5 m and the mass was pulled 12.6 cm away from its equilibrium position. Therefore, refer to the diagram below, L = 1.5 m and x = 0.126 m.

In the diagram,

        Refer to the displacement-acceleration (x-a) graph, the line plotted is a = -constant x + c where constant is 2.3 and c is 0.7. The line does not pass the origin but cut the y-axis. It is different from the theoretical one a = -constant x. However, this will not affect the slope of the line. The angular velocity (ω) found will not be affected. It will only affect the y-intercept of the graph.

In the above graph, , the negative sign indicates that acceleration is always at an opposite direction to displacement and is directed to the equilibrium position.

a = -constant x indicates that acceleration is directly proportional to displacement.

Also consider the constant in the theoretical graph and the experimental graph.

The theoretical one,

constant =

The experimental one,

constant = slope of line =2.3

There is a large difference between the two values. The percentage error is

The reason for such a large percentage error will be discussed in ‘Error analysis’ of this report.

Improvement

• It is better to use a computer to detect the motion and to analyze the result obtained. Analyzing the data by hand produces a large error.
• We should be sure that our hands do not give an external force to the pendulum bob when releasing it.
• The experiment can be done in vacuum to avoid the effect of air resistance.
• In fact, it is impossible to eliminate the friction between the ticker-tape and the ticker-tape timer. The only thing we can do is to be sure that the oscillation and the ticker tape timer are on the same plane. This can minimize the friction.
• We have improved the overlapping of dots during the experiment by switching off the ticker-tape timer immediately after the pendulum bob reached its highest point.

Conclusion

From this experiment, we know that the acceleration and displacement of a simple harmonic motion is directly proportional to each other. Also, the acceleration is always opposite to displacement from the equilibrium position.

However the experiment just shows that acceleration and displacement is directly proportional to each other and acceleration is always opposite to displacement from the equilibrium position. It cannot show the exact relationship between them. ()

To conclude, I am not so satisfy with the result obtained in this experiment, as the error is very large.

Displacement-time (x-t) graph

Velocity-time (v-t) graph

Displacement-acceleration (x-a) graph