A simple pendulum is suspended from the ceiling a car accelerating uniformly on a horizontal road. If the acceleration is `a_0` and the length of the pendulum is l, find the time period of small oscillations about the mean position.

Explanation

We shall work in the car frame. As it is accelerated with respect to the road, we shall have to apply a pseudo force `ma_(0)` on the bob of mass `m`.
For mean position, the acceleration of the bob with respect to the car should be zero. If `theta_(0)` be the angle made by the string with the vertical, the tension, weight and the pseudo force will add to zero in this position.
Hence, resultant of `mg` and `ma_(0)` (say `F = msqrt(g^(2) + a_(0)^(2))`) has to be along the sting.
`:. tan theta_(0) = (ma_(0))/(mg) = (a_(0))/(g)`
Now, suppose the string is further deflected by an angele `theta` as shown in figure.
Now, restoring torque can be given by
`(F sintheta)l = -ml^(2)alpha`
Substituting `F` and using `sintheta = theta`, for small `theta`.
`(msqrt(g^(2) + a_(0)^(2))) l theta = -ml^(2)alpha`
or, `alpha = - sqrt(g^(2) + a_(0)^(2))/(l)theta` so, `omega = sqrt(g^(2) + a_(0)^(2))/(l)`
This is an equation of simple harmonic motin with time period
`T = (2pi)/(omega) = 2pi(sqrt(l))/(g^(2) + a_(0)^(2))^(1//4)`