Question

A car is speeding up on a horizontal road with an acceleration $\alpha$. 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. 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 the horizontal.

Answer 1

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

Let $m$ be the mass of the pendulum

From the free body diagram

$T\cos \theta -mg=0$

$T\cos \theta =mg$

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

$T\sin \theta -ma=0$

$T\sin \theta =ma$

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

From (i) and (ii)

$\frac{mg}{\cos \theta }=\frac{ma}{\sin \theta }$

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

$\theta ={\tan }^{-1}\frac{a}{g}$

Thus angle is ${\tan }^{-1}\frac{a}{g}$ with vertical

(ii) Suppose the angle of incline is $\theta$

From the diagram

$ma\cos \theta -mg\sin \theta =0$

$ma\cos \theta =mg\sin \theta$

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

$\theta ={\tan }^{-1}\frac{a}{g}$