Question

A ball is suspended on a thread from the ceiling of a car. The brakes are applied and the speed of car changes from $5\, m/sec$ to $5/3 \,m/sec$ during the time interval of $3$ seconds. Find the angle that the thread with the ball will deviate from vertical

• A. $\theta = tan^{-1} \left(\frac{1}{9}\right)$
• B. $\theta = tan^{-1} \left(\frac{8}{9}\right)$
• C. $\theta = sin^{-1}\left(\frac{1}{9}\right)$
• D. $\theta = cos^{-1}\left(\frac{1}{9}\right)$

Answer 1

Answer: (A)

The situation is shown in figure. When the car moves towards right with acceleration a the rope carrying the mass makes an angle $\theta$
Forces acting on the body are:
(1) Weight mg downwards
(2) Tension T in the rope
(3) Pseudo force ma towards left
Since the body is in equilibrium under the action of three forces, we can apply condition of equilibrium
$T\,cos\,\theta-mg=0 ; T\,sin\,\theta=ma$
$tan\,\theta=\frac{a}{g}$ or $a=g\,tan\,\theta$
Acceleration, $a=\frac{1}{3}\left(5-\frac{5}{3}\right)$
$=\frac{10}{9}m/sec^{2}$
$\frac{10}{9}=10\,tan\,\theta$ or
$tan\,\theta=\frac{1}{9}$
$\Rightarrow \theta=tan^{-1} (\frac{1}{9})$