A simple pendulum of length l is suspended from the ceiling of a...

In the frame of the car, the pendulum bob experiences a pseudo force (centrifugal force) in the horizontal direction due to the circular motion of the car.

The centrifugal acceleration is $a_c = \dfrac{v^2}{r}$ directed radially outward.

The acceleration due to gravity $g$ acts vertically downward.

Since the horizontal and vertical accelerations are perpendicular to each other, the effective acceleration due to gravity $g_{eff}$ is given by:

$g_{eff} = \sqrt{g^2 + a_c^2} = \sqrt{g^2 + \left(\dfrac{v^2}{r}\right)^2} = \left( g^2 + \dfrac{v^4}{r^2} \right)^{1/2}$

The time period of small oscillations of a simple pendulum is given by:

$T = 2\pi \sqrt{\dfrac{l}{g_{eff}}}$

Substituting the value of $g_{eff}$, we get:

$T = 2\pi \sqrt{\dfrac{l}{\left( g^2 + \dfrac{v^4}{r^2} \right)^{1/2}}}