### Simple Pendulum in Inertial Frame:

A heavy point mass (bob), suspended by a light, long and inextensible string, forms a simple pendulum.

Length of the simple pendulum is the distance between the point of suspension and the centre of mass of the suspended mass.

Consider the bob when string deflects through a small angle θ from equilibrium position.

Forces acting on the bob are tension (T) in the string and weight (mg) of the bob.

Torque on the bob about point O is

τ = τ_{mg} + τ_{T} = mg l sinθ + 0

τ = mg l θ (as θ is very small) …(1)

Moment of inertia of the bob about the point O is I = m l^{2}

$ \displaystyle \tau = I\alpha = ml^2\frac{d^2\theta}{dt^2} $ …(2)

For anticlockwise θ , τ is clockwise, hence from (1) and (2), we get

$ \displaystyle ml^2\frac{d^2\theta}{dt^2} = -mgl\theta $

$ \displaystyle \frac{d^2\theta}{dt^2} = -\frac{g}{l}\theta $

Comparing with the equation $ \displaystyle \frac{d^2\theta}{dt^2} = -\omega^2 \theta $ , we get

$ \displaystyle \omega = \sqrt{\frac{g}{l}} $

Since ,

$ \displaystyle T = \frac{2\pi}{\omega} $

$ \displaystyle T = 2\pi \sqrt{\frac{l}{g}} $

Above result is derived using the concept of torque. The same can be derived using the concept of force also.

__Simple Pendulum in Non Inertial Frame__

If a simple pendulum is made to oscillate in a non inertial frame the pseudo force must or torque due to pseudo force should also be taken into account. For a simple pendulum inside a frame accelerating with acceleration $\vec{a}$ with respect to an inertial frame, time period is given by

$ \displaystyle T = 2\pi \sqrt{\frac{l}{|\vec{g}-\vec{a}|}} $

For example,

(i) Time period of pendulum inside a cart moving with acceleration ‘ a ‘ on horizontal road

$ \displaystyle T = 2\pi \sqrt{\frac{l}{\sqrt{g^2 + a^2}}} $

(ii) Time period of the pendulum inside an elevator accelerating in upward direction with acceleration a

$ \displaystyle T = 2\pi \sqrt{\frac{l}{g + a}} $

(iii) Time period of the pendulum inside an elevator accelerating in downward direction with acceleration a

$ \displaystyle T = 2\pi \sqrt{\frac{l}{g – a}} $

Note: If acceleration of non-inertial frame with respect to inertial frame is not in vertical direction then in equilibrium position, the string of pendulum makes some angle with vertical.

Exercise : Find the time period of a simple pendulum of length l suspended from the ceiling of a car moving with a speed v on a circular horizontal rod of radius r.

Exercise : Find the time period of a simple pendulum of length L having a charge q on its bob when the pendulum is oscillating in a uniform electric field E directed (a) parallel to g (b) perpendicular to g

Exercise : A hollow metal sphere is filled with water and a small hole is made at its bottom. It is hanging by a long thread and is made to oscillate. How will the time period change if water is allowed to flow through the hole till the sphere is empty.