Question

A simple pendulum of length l is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation.

Answer 1

Explanation ⇒ When the car is moving in an Circular path, then it will experiences an Centripetal force acting towards the centre of circular path.

But, Pendulum kept inside will be in non-inertial frame, thus, It will experience as centrifugal force outside.

Force of gravity will be acting downwards for pendulum bob of mass m.

Refer to the attachment.

(a).  In Equilibrium,

TSinθ = mv²/r   ---(I)

TCosθ = mg.    -----(2)

∴ (I)² + (2)²

∴ T²(Sin²θ Cos²θ) = m²v⁴/g² + m²g²

∴ T²  = m²v⁴/g² + m²g²

∴ T = √(m²v⁴/g² + m²g²)

Hence, the force of tension is √(m²v⁴/g² + m²g²).

(b). ∴ Net Force = √[(mv²/r)² + (mg)²]

∴ a = √[v⁴/r² + g²]

Now, using the formula,

$T = 2\pi \sqrt{\frac{l}{a _e_f_f} }$

$T = 2\pi \sqrt{\frac{l}{\frac{v^4}{r^2} + g^2 } }$

This is the required time period of the Pendulum.

   

Hope it helps.