Question

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 angle with the vertical and does not slip on the incline. Find the angle of the incline with the horizontal.

Answer 1

Suppose pendulum makes angle $\theta$ with the vertical.

Let m = mass of the pendulum

From the free body diagram,

$Tcos\theta -mg=0$

$\Rightarrow Tcos\theta =mg$

$\Rightarrow$ T = $\frac{mg}{cos\theta }$ ...(i)

and $ma-Tsin\theta =0$

$\Rightarrow ma=Tsin\theta$

$\Rightarrow T=\frac{ma}{sin\theta }$ ....(ii)

$\Rightarrow tan\theta =\frac{a}{g}$

$\Rightarrow \theta =ta{n}^{-1}\frac{a}{g}$

i.e. angle formed is $ta{n}^{-1}\left(\frac{a}{g}\right)$ with vertical.

(ii) Let M = mass of bloc

Suppose the angle of incline is $\theta$,

From the diagram,

$\Rightarrow macos\theta =mgsin\theta$

$\frac{sin\theta }{cos\theta }=\frac{a}{g}$

$tan\theta =\frac{a}{g}\theta =ta{n}^{-1}\left(\frac{a}{g}\right)$

The angle of incline is $ta{n}^{-1}\left(\frac{a}{g}\right)$