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Investigate the factors which determine the damping of a compound pendulum to find an equation that relates the amplitude of oscillations to the factors chosen to investigate.

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Introduction

Physics A2 Assessed Practical

Aim: Investigate the factors which determine the damping of a compound pendulum to find an equation that relates the amplitude of oscillations to the factors chosen to investigate.  

Compound Pendulum

For a system to oscillate in simple harmonic motion there are 3 conditions which should be satisfied;

1. A mass that oscillates,

2. A central point where the mass is in equilibrium,

3. A restoring force which returns the mass to its central point.

The compound pendulum (shown above) clearly does oscillate with S.H.M as there is a mass that oscillates (1) about an equilibrium point (2) and a restoring force returning it to its central point (weight of the mass / tension in ruler (3)).

In S.H.M, there is a constant interchange between kinetic and potential energy. In the case of the compound pendulum the potential energy is provided by the increase in gravitational potential energy (mgΔh) as the oscillations occur in a circular fashion taking the mass higher above the ground at its maximum / minimum displacements. So in an ideal situation (one where 100% of the ΔEp is converted to kinetic energy) the oscillations should go on forever with constant maximum / minimum amplitudes.  


image00.png

However, we can see in our everyday lives, in such situations as a car suspension system or a child on a swing, that this is not the case.

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Middle

I also decided that friction at the pivot of the system would have an effect on the damping. I should aim to reduce this factor as much as possible as it will inevitably create errors in my readings. I found during my preliminary experimentation that placing a washer between the pivot and the clamp did help reduce the friction but I obviously will never totally decrease the friction to nothing.

Another factor I need to consider is the position of the card on the ruler. Firstly I decided to attach one piece of card to just one side of the ruler. Although only the leading edge of the pendulum has any significant effect on the damping of the pendulum, to keep the test simple (so as not to bring any other variables into consideration) I felt it best to just use one card.

I then performed some preliminary investigation to decided how far up the ruler the card should be attached.

Centre of card attached

xm above mass

Time for ½Ao (s)

0.20

10.08

0.40

10.32

0.60

10.98

Effective length = 0.80m   Mass = 0.200kg

Surface area of card = 0.21m2

The distance up the ruler had surprisingly little effect on the damping but seen as attaching the card 0.20m above the mass had the greatest effect I will use that figure in my experiment.  


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Conclusion

most probably not be linear. At first when the system is moving quickly there will be a lot of air resistance to overcome so energy will lost quickly. As the system slows down there will be less air resistance so energy will be lost at a slower rate as shown in the graph below.

image01.png

The relationship will be in the form T = kAn where T is the time taken for ½Ao, A is the surface area of card attached to the pendulum and k and n are constants (n should work out to be a negative number to represent the inverse proportionality).

To find values for k and n I will need to plot a straight line graph and compare my relationship to the equation y = mx + c. To do this I will have to take logs of the equation T = kAn so as to turn the power n into a coefficient of A.

So T = kAn becomes logT = nlogA + logk which compares very easily to the equation for a straight line;

                                                y     =   m     x       +     c

                                             logT   =   n   logA    +   logk

This shows that if I plot logT on the y axis and logA on the x axis, I should get a straight line with gradient n intercepting the y axis at logk.


Results

Surface area of card (m2)

Time for ½Ao (s)

Average time (s)

0.021

9.98

10.25

10.44

10.32

0.0175

11.04

11.48

11.56

11.84

0.014

12.60

12.88

12.89

13.16

0.0105

16.41

16.67

16.46

17.15

0.007

23.20

23.56

23.99

23.50

0.0035

33.37

34.14

34.97

34.07

LogT

1.01

1.06

1.11

1.22

1.37

1.53

LogA

-1.68

-1.76

-1.85

-1.98

-2.15

-2.46

...read more.

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