SIMPLE PENDULUM

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By Aditya Abeysinghe

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A simple pendulum when given a small displacement obeys simple harmonic motion.

Thus, the energy conversion is as follows:

1. Maximum kinetic energy at the base

2. Maximum potential energy at the amplitude

3. Conservation of mechanical energy at any point in its motion

Now let’s examine as to which factors affect the motion of a simple pendulum.

PRINCIPLE BEHIND A SIMPLE PENDULUM

It should be noted that the tension is balanced by the component of

weight mg cosθ.

The other component of the weight mg sinθ is used for the acceleration of the object.

By newton’s second law of motion,

F = ma

F = -mg sinθ

Since θ is small, sinθ ≈ θ rad. Therefore, F = -mgθ

θ

T

mg

l

x

However, S = rθ and hence x = lθ. Therefore, θ = x /l

Thus, F = -mgx/l

By newton’s second law,

F = ma

Therefore, ma = -mgx/l and a = - (g/l) x

This is in the form a = -ω2x.

Therefore, the object obeys simple harmonic motion.

And ω = g/l.

However, T = 2π/ω

Finally, we get T = 2π √ (l / g)

Moreover, we can write the equation as follows:

T = 2π √ (l / g)

By squaring both sides,

T2 = (4π2 / g) l

This is in the form y = mx

Where the gradient of the graph = 4π2 / g

By performing the experiment and plotting a graph, we can easily find the gravitational acceleration using the gradient.

Required materials:

• Simple pendulum

• Stop watch

• Meter ruler

• Stand with a fixed pointer

• A weightless string

THE EXPERIMENT

1. Place the apparatus and keep the length of the pendulum to be 2m.

2. Place the pointer close to the lowest point of the string

3. Displace the string either clockwise or counterclockwise/anticlockwise and then release the string

4. At the first instance when the string passes the pointer the stop watch is activated.

5. Take the readings to 50 complete revolutions.

6. Repeat the above experiement if the difference between the time periods of revolutions is greater than 0.5s

7. Finally, decrease the length of the string by 0.25m gradually and draw the graph between l and T2

METHOD

1. Use a string since when the string is in motion, the increase the length of the string

2. To prevent changing the length of the string, the string should be swayed perpendicular to the stopper, to where the string is connected above.

3. The length of the string, l, should be measured from the center of gravity of the weight to the stopper above.

4. By keeping the pointer at the lowest point of the string’s path of motion, the time can be accurately measured since the string sways at its highest speed at the lowest point

IMPORTANT POINTS