A simple pendulum of length l and mass m is suspended from the...

In the frame of reference of the car, the forces acting on the pendulum bob are the gravitational force $mg$ acting vertically downwards, the centrifugal force $\dfrac{mv^2}{r}$ acting horizontally outwards, and the tension $T$ in the string.

Since the pendulum is at rest with respect to the car, it is in equilibrium in the car's frame. The tension $T$ in the string must balance the resultant of the gravitational force and the centrifugal force.

As the gravitational force and the centrifugal force are perpendicular to each other, their resultant is given by:

$F_{res} = \sqrt{(mg)^2 + \left(\dfrac{mv^2}{r}\right)^2}$

Therefore, the tension in the string is:

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