• 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.  

...read more.

Middle

image26.png.

Dividing both sides by m and using the small angle approximationimage27.png, the equation of motion is

image28.png.

This can alternatively be written as

image29.png  or   image30.png

        In this case because image31.png 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 image31.png by x. The equation will looks like this: image33.png or image34.png. From this equation, we know that image35.png. In this particular coursework the length of the string I use is 1 meter long, hence image36.png where g is 9.8.

        The damped term image37.png 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.image38.png

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

...read more.

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 image47.png. The general solution of the differential equation is image51.png and the particular solution is image57.png. 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

...read more.

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

See related essaysSee related essays

Related GCSE Forces and Motion essays

  1. Period of Oscillation of a Simple Pendulum

    I shall also make comparisons to the theoretical answers put forward by Galileo * Length of pendulum Here, the results are very good, with no anomalous, unexpected results. The results that I had were very accurate except for the lengths of 20 cm and 65 cm, which both suffered the worse scores compared to Galileo's theoretical answer.

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

    Manually timing the experiment: Added weight (g) Time taken to travel 2m (s) Velocity [distance/time] (m/s) Average speed (m/s) 0 3.51 3.44 3.32 0.64 0.58 0.61 0.61 200 2.33 2.17 2.13 0.86 0.92 0.94 0.91 400 2.26 2.15 2 0.88 0.93 1 0.94 600 2 2.15 2.16 1 0.93 0.93

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

    The pendulum would also be checked before any reading to make sure that it is hanging correctly before every reading. The wooden blocks need to be kept tight, to ensure that the pendulum is only swinging from one fixed point.

  2. Investigating the amazingness of theBouncing Ball!

    amount of kintetic energy and therefore does not reach its original height. Howerver the relative velocity with which two bodies separate from each other, after a collion, is related to their relative velocity of approach and a constant known as the 'coefficient of resitution'of the two bodies.

  1. report on glass

    AD 100 The first people to use glass for an architectural purpose were the Romans. With the discovery of clear glass through the introduction of manganese oxide. Cast glass windows with poor optical qualities, then began to appear in the most important buildings in Rome.

  2. 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

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

    By increasing the length of the string, one side of the arc i.e. the radius increases and hence the object has to travel through a larger arc and a greater distance provided that the mass of the object and the angle remain constant.

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

    be cut precisely whereas the fishing line idea has a round bob with no definite cut of point and the fishing line is so thin it would not cut the light. As long as the cut of point is the same each time a light gate should be very accurate

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