A pendulum is hanging from the ceiling of a car having an acceleration `a_0` with respect to the road. Findthe angle made by the string with the vertical.

Explanation

The situation is shwon in ure. Suppose themss of the bob is m and the string makes an angel `theta` with the vertical. We shall work from the car frame. This frame is noninertial as it has an acceleration `veca_0` with respect to an inretial fme (the road). Hence, if we suse Newton's second law shal have toinclude pseudo force.

Take the bob as the system.
The force are:
a. T along the string, by the string
b. mg downward, by the earth
c.`ma_0` towards left (pseudo force).
The free body diagram is shown in ure. As the bob is at rest (remember we are discussing the motion with respect to the car) the force is a,b, c should add to zero. Take X-axis along the forward horizontal direction and Y-axis along the upward vertical direction. The components of the along the X-axis gie
`T sin theta -ma_0=0 or, T sin theta = ma_0` ...........i
And the components along the Y-axis give
`Tcos theta -mg=0 or, T cos theta=mg` ..........ii.
Dividing i. by ii `tantheta =a_0/g`
Thus, the string makes an angle `tan^-1(a_0/g)` with te vertical.