Question

A car of mass m is moving on a level circular track of radius R.If $\mu_s$ represents the static friction between the road and tyres of the car, the maximum speed of the car in circular motion is given by

• A. $\sqrt{\mu_s mRg}$
• B. 0
• C. $\sqrt{\frac{mRg}{\mu_s}}$
• D. $\sqrt{\mu_s Rg}$

Answer 1

The Correct Option is D

To determine the maximum speed of the car when moving on a level circular track, we need to understand the forces acting on the car. The maximum speed of a car moving in circular motion without skidding is primarily determined by the frictional force acting between the tyres and the road.

Here, the static friction provides the necessary centripetal force to keep the car in circular motion. Thus, we can establish the equation:

F_{\text{centripetal}} = F_{\text{friction}}

Where

• F_{\text{centripetal}} = \frac{mv^2}{R} is the centripetal force required to keep the car moving in a circle with speed v and radius R.
• F_{\text{friction}} = \mu_s mg is the maximum static frictional force, with \mu_s being the coefficient of static friction, m the mass of the car, and g the acceleration due to gravity.

Equating these forces, we have:

\frac{mv^2}{R} = \mu_s mg

We can cancel m from both sides as it appears non-zero:

\frac{v^2}{R} = \mu_s g

Solving for v (the maximum speed), we get:

v^2 = \mu_s g R

Taking the square root of both sides gives:

v = \sqrt{\mu_s g R}

Therefore, the maximum speed of the car in circular motion is \sqrt{\mu_s g R}.

By comparing with the given options, the correct answer is \sqrt{\mu_s R g}.