# Studying a simple harmonic oscillator.

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Introduction

Name: Leung Hoi Ning (14) Class: 6S Group No.: 8 Date of experiment: 9-12-03

Experiment B

Title Studying a simple harmonic oscillator

Objective We are studying the simple harmonic motion of a pendulum by attaching a ticker-tape to the bob and analyzing the dots marked on the tape.

Experimental Design

- Apparatus: 0.5 kg ringed mass

1.5 m length of string

Ticker-tape timer

Ticker-tape

Low voltage power supply (a.c.)

Retort stand and clamp

- Procedure for getting the ticker-tape

- A string with 1.5 m long was measured and tied with the 0.5 kg ringed mass
- Set the pendulum as shown in figure 1

Figure 1

3. The ticker-tape timer was connected to the low voltage power supple (a.c.)

- The retort clamp was used to hold the clamp tightly so it would not vibrate when the mass was swinging.
- A 30cm ticker-tape was attached to the ringed mass
- The 0.5 kg ringed mass was pulled to one side with amplitude of 13cm from the equilibrium position.
- The power supply and the ticker-tape timer were switched on.
- The 0.5 kg ringed mass was allowed to swing to the other side after a few dots were stroke on the same place of the ticker-tape.
- The ticker-tape timer was switched off when the 0.5 kg ringed mass began to swing back to the equilibrium position.
- Step (3) to step(7) were repeated until 5 more ticker-tapes were got.

- Procedures for plotting graphs

- The dots marked on the tape were examined.

Middle

0.088

0.065

0.046

0

-0.029

-0.059

-0.083

-0.102

-0.117

-0.124

Acceleration a/ cm/s² | -1.55 | -1.43 | -1.25 | -0.95 | -0.65 | 0 | 0.4 | 0.8 | 1.15 | 1.45 | 1.65 | 1.75 | |||||||||||

Time t/s | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.1 | 1.2 |

Data evaluation:

From the displacement-time graph, the curve is a half cycled cosine curve. The curve in the velocity-time graph is a negative, half-cycled sine curve. The curve in the acceleration-time graph is a negative, half-cycled cosine curve. From the acceleration-displacement graph, a straight line was shown, which the acceleration is always in the opposite direction and is directly proportional to the displacement. So we can use the following equation to represent the motion:

a = -w ² x

(where a is the acceleration, w is the angular velocity ( a positive constant ), x is the displacement)

The definition of simple harmonic motion (S.H.M.) is:

- the acceleration of a particle is

- directed towards a fixed point
- directly proportional to its distance from that poing

- the acceleration is always in the opposite direction to the displacement

So, the equation a = -w²x proved that the pendulum performed S.H.M.

Also, for a S.H.M. the angular velocity (w) is kept constant. Period (T) is equal to 2∏/w, so the period is also kept constant and is independent of amplitude and mass of the bob.

From the graphs the value of w is more or less the same within the motion.

time | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.1 | 1.2 |

w | 0.007 | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.01 | 0.03 |

As period (T) = 2∏ / w

Conclusion

The swing angle is difficult to keep under 5 º. So if an angle larger then 5 º is used, the equations

cannot be held and so the motion is not a simple harmonic one.

When leave the ringed mass and let it to swing, we may give an external force to it. Also the mass oscillates horizontally during the swinging motion. These may affect the swing angle and the velocity of the bob and this explained why the dots on the v-t graph cannot connect together to form a smooth curve.

Another big problem is to measure the slopes of the curves accurately. This explains why the dots in v-t, x-a and a-t graphs cannot connect together to form smooth curves and straight line.

Improvements

- Hold the mass stationary for a while before leaving it. Try not to exert any force to the mass.
- Twist the clamp as tight as possible so the whole system will not vibrate or oscillate.
- Set the ticker-tape timer at a place which cannot touch the chair, while keeping on the same horizontal plane with the mass.
- use a plane mirror to get the normal of the dots and then calculate the slopes of the dots.

Conclusions

Simple pendulum is a kind of simple harmonic oscillator. The experimental results satisfied the equation for simple harmonic motion ( a = - wx)

This student written piece of work is one of many that can be found in our GCSE Forces and Motion section.

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