A car is speeding up on a horizontal road with an acceleration a. Consider the following situations in the car. i. A ball is suspended from the ceiling through a string and is maintaining a constant angel with the vertical. Find this angle. Ii. A block is kept on a smooth incline and does not slip on the incline. Find the angle of the incline with a horizontal.

Answer

A, C

Explanation

Suppose pendulum makes angle theta with the vertical.
Letm=mass of the pendulum
From the freee body diagram
`T cos theta - mg = 0`
` rarr T cos theta = mg`

`rarr T= (mg) / (cos theta) ` …....i
` and ma-T sin theta =0`
` rarr ma= T sin theta`
` rarr T= (ma)/(sin theta)` ..........ii
`rarr tan theta = a/g`
` rarr theta= tan^-1 = a/g`
i.e. angle formed is `tan^-1 (a/g)` with verticle.
ii. Let M mass of block
Suppose the ngle of incline is theta,
From the diagram,
`rarr ma cos theta = mg sin theta`
` (sin theta)/(cos theta)=a/g`
` tan theta = a/g theta = tan^-1 (a/g)`
The angle of incline is `tan^-1 (a/g)`.