Question

A car is speeding up a horizontal road with acceleration a. Consider the following situations in the car: (i) A ball suspended from the ceiling by a string 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

Let the pendulum (formed by the ball and the string) make angle θ with the vertical.


From the free-body diagram,
Tcosθ − mg = 0
Tcosθ = mg
$\Rightarrow T=\frac{mg}{\cos \mathrm{\theta }}...\left(\mathrm{i}\right)$
And, ma − T sin θ = 0
⇒ ma = T sin θ
$$\begin{gathered} \Rightarrow T=\frac{ma}{\sin \mathrm{\theta }}....\left(\mathrm{ii}\right) \\ \Rightarrow \tan \mathrm{\theta }=\frac{a}{g} \\ \Rightarrow \mathrm{\theta }={\tan }^{-1}\frac{a}{g} \end{gathered}$$
So, the angle formed by the ball with the vertical is ${\tan }^{-1}\left(\frac{a}{g}\right)$ .

(ii) Let the angle of the incline be θ.
From the diagram,


⇒ ma cos θ = mg sin θ
$$\begin{gathered} \frac{\sin \mathrm{\theta }}{\cos \mathrm{\theta }}=\frac{a}{g} \\ \tan \mathrm{\theta }=\frac{a}{g}\mathrm{\theta }={\tan }^{-1}\left(\frac{a}{g}\right) \end{gathered}$$
So, the angle of incline is ${\tan }^{-1}\left(\frac{a}{g}\right)$.