• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12

# Damped Oscillation.

Extracts from this document...

Introduction

MECHANICS 4

Coursework

Work based on a combination of the Modelling and Experimental cycles

Damped Oscillation

By Jian Qin Lu

• Introduction

Simple Harmonic Motion (SHM) is a very interesting motion. In the ideal situation the acceleration of the moving object is proportional to the distance between the object and the origin (O), and the time period of the oscillation is constant. However, in a real situation the motion doesn’t exactly behave like this. Because there is damping, it makes the amplitude of the motion decrease and finally the motion will be stopped. The whole system continues lose energy due to against the resistance (i.e. air resistance).

Simple pendulum motion can be approximated as a SHM at a small angle (less than 170 or 0.3 radius at 2 decimal places accuracy level). Therefore it can be modelled as SHM (with the damping term).

(NB: all the time measurements in this coursework are accurate to 0.01 second.)

• Aim

In this coursework I am going to use differential equation to model the damping of the simple pendulum motion in a thin liquid. And find the general solution of the differential equation. Also I will give the particular solution of this situation.

• Simplifying the situation and setting up the model

Here I will list the basic data of the experiment.

1. The mass of the pendulum bob (m) = 1kg
2. The length of the string (l) = 1m

3.

Middle

.

Dividing both sides by m and using the small angle approximation, the equation of motion is

.

This can alternatively be written as

or

In this case because  is the only factor which can vary x which is the displacement from centre to the current pendulum bob location (length of the string is constant). Hence I can replace  by x. The equation will looks like this:  or . From this equation, we know that . In this particular coursework the length of the string I use is 1 meter long, hence  where g is 9.8.

The damped term  in fact is the resistance. I did a separate experiment in order to find the resistance.

What I did is that dropped the pendulum bob into a long tube full of water from the surface of water. I made a mark on the tube whose position is 0.2 meter from the surface of water. I would like to record the time which the ball took to cover this distance. In this experiment I assumed that the acceleration in this period is constant and the ball did not reach its terminal velocity in the 0.2 meter distance. The left hand side diagram shows the set of the experiment.

In order to reduce the error, I just took the reading from the first 0.2 meter and I repeated this experiment twice.

Conclusion

Anyhow in the experiment I did, my model worked. It can represent the motion pretty well. Therefore I consider that this differential equation is the model of this situation.

• Assessment of the improvement

Since my reaction time is the biggest factor which effect on the experiment, I could use some better equipment to record the time for me. The reading will be more accurate than that I gained.

Also I can consider more factors rather than make assumptions. The model should be able to represent the real situation better.

• Conclusion

Through this coursework I have obtained the differential equation which models the motion of a 1kg bob with 1 meter string in the water. The differential equation is . The general solution of the differential equation is  and the particular solution is . The whole is system is overdamping.

The biggest variability in my coursework is the time measurements. It may change the parameter of the damped term. Therefore the type of damping may be changed as well. However according to the experiment, the whole system seems cannot be underdamping.

• Reference

1. Differential Equations by Mike Jones and Roger Porkess

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Forces and Motion essays

1. ## Period of Oscillation of a Simple Pendulum

I have drawn a scatter-graph to show these results. The scatter-graph shows the results of my experiment (in red) and the theoretical results (in green). The green dots at which the red dot cannot be seen, have the best results as this shows that the results are close to the theoretical answer.

2. ## Determining the acceleration due to gravity by using simple pendulum.

We have neglected the air resistance, mass and many other variables which could help to find the accurate value of acceleration due to gravity. The sources of errors in my experiment were rounding off values which may vary the outcome of the results calculate with the formula.

1. ## In this experiment I aim to find out how the force and mass affect ...

desk exerts a friction force in the opposite direction of its motion. Friction results from the two surfaces being pressed together closely, causing intermolecular attractive forces between molecules of different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together.

2. ## The determination of the acceleration due to gravity at the surface of the earth, ...

It is necessary to make sure that the wooden blocks have well-defined right angles and are clamped in well. The string of the pendulum should hang at 90� when in equilibrium position. This measurement needs to be checked before any readings or results are taken.

1. ## In this Coursework, we were given the task of investigating some factors which affect ...

Now that I have explained how the apparatus was set up, let me go on to explain the procedure for each experiment. EXPERIMENT NO 1: VARYING THE LENGTH The following provides a step by step guide as to how this experiment was carried out: 1- 1 metre length of string

2. ## Investigating the amazingness of theBouncing Ball!

Type of collision e Elastic 1 Inelastic <1 Completly inelastic 0 Having conducted the method established using a simple ping pong ball, I will apply my results to the formula to find the ping pong's coefficient of restitution. So from my results the speed of approach before the first bounce ie.

1. ## Mechanical Properties of a Meter Rule

With this experiment I will be able to find many different things, like the modulus of elasticity for different materials, bending stress, the energy stored as it deforms etc. . The second experiment I will do is the compound pendulum, with this I will be able to work out the

2. ## An Experiment Using a Pendulum to Find the Acceleration due to Gravity.

Rigid pendulums are used in clocks so they must be accurate as timekeepers. Yet an Internet site (http: kossi.physics.home.edu/Courses/p23a/Experiaments/Pendulum.html) about the experiment stated that it recommended the use of a massless, inextensible string. All experiments I have seen also use some sort of string rather than a rigid structure.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to