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.

#### Solution 1

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

*F* = –*mg* sin*θ*

Where,

*F *= Restoring force

*m* = Mass of the bob

*g* = Acceleration due to gravity

*θ* = Angle of displacement

For small* θ*, sin*θ* = *θ*

For large* θ*, sin*θ* 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

#### Solution 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)`