Question

A man with his hands in his pocket is standing in a bus , which is taking a turn in a horizontal curve of radius 50 m , with a speed of 7 m/s far from the vertical must he lean to keep his balance ?
(A) $\theta = {\tan ^{ - 1}}(0.2)$
(B) $\theta = {\tan ^{ - 1}}(0.15)$
(C) $\theta = {\tan ^{ - 1}}(0.1)$
(D) $\theta = {\tan ^{ - 1}}(0.05)$

Answer 1

Hint : Motion along the curved path is different from that on the straight line. The motion along the curved path cannot actually be the same as that of the circular motion. But, as we proceed they can be analyzed by using the dynamics of circular motion.
The change in posture along the curved path can be seen.
The angle $\tan \theta = \dfrac{{{v^2}}}{{rg}}$
v: speed of the vehicle
r: radius of the curved path
g: acceleration due to gravity.

Complete Step By Step Answer:
In this question we are provided with speed of vehicle along the curved path
$v = 7m/s$
The radius of curvature $= 50m$
It is mentioned the vehicle is taking a sharp turn along the circular path.
If a person is sitting in the middle of the backseat of the vehicle in our case the bus he will maintain the posture continuously. But, if sitting nearby or far from the center of the curve will experience a force.
This is centripetal force acting on the body. The posture of the body will change along with the motion of the vehicle along the curved path.
When the curved oath is horizontal this angle at which we lean will be dependent on the velocity of the bus and radius of the curvature.
The formula is given by,
$\tan \theta = \dfrac{{{v^2}}}{{rg}}$
v: speed of the vehicle
r: radius of the curved path
g: acceleration due to gravity
substituting all the values in the equation we will get,
$\tan \theta = \dfrac{{{7^2}}}{{50 \times 9.8}} \\\ \Rightarrow \tan \theta = \dfrac{{49}}{{490}}$
By further solving and carrying out the division we get
$tan\theta = \dfrac{1}{{10}} \\\ \Rightarrow \tan \theta = 0.1 \\\ \Rightarrow \theta = {\tan ^{ - 1}}(0.1)$
Hence option (C) is correct.

Note :
There are three aspects to negotiating a curve.

1. We want to travel at high speed along the curve
   2)If we want to drive safely.
2. We want to prolong the life of tires by avoiding sideways friction.
   This may be achieved by banking the roads in which angle of banking is $\theta = {\tan ^{ - 1}}\left( {\dfrac{{{v^2}}}{{rg}}} \right)$ .