Question

Answer the following questions:

The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than `2pisqrt(1/g)`  Think of a qualitative argument to appreciate this result.

Answer 1

In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as:

F = –mg sinθ

Where,

F = Restoring force

m = Mass of the bob

g = Acceleration due to gravity

θ = Angle of displacement

For small θ, sinθ = θ

For large θ, sinθ is greater than θ.

This decreases the effective value of g.

Hence, the time period increases as:

`T = 2pi sqrt(1/g)`

Where, l is the length of the simple pendulum

Answer 2

The restoring force for the bob of the pendulum is given by

`F = -mg sintheta`

if  `theta` is small thensin `theta = theta = y/l` `:. F = -(mg)/l y`

i.e the motion is simple harmonic and time period is` T = 2pi sqrt(1/g)`

Clearly, the above formula is obtained only if we apply the approximation `sin theta ~~ theta`

For large angles this approximation is not valid and T is greater than `2pi sqrt(1/g)`