Question

A car of mass m is moving on a level circular track of radius R. If μs represents the static friction between the road and tires of the car, the maximum speed of the car in a circular motion is given by:
1. \(\sqrt {{\mu _s}mRg}\)
2. \(\sqrt {Rg\;/\;{\mu _s}}\)
3. \(\sqrt {mRg\;/\;{\mu _s}}\)
4. \(\sqrt {{\mu _s}Rg}\)

Answer 1

Correct Answer - Option 4 : \(\sqrt {{\mu _s}Rg}\)

CONCEPT:

• Centripetal force(F_c): It is the force that is necessary to keep an object moving in a curved path and that is directed inward toward the center of rotation.

  • For example, the tension in the rope on a tetherball, the force of Earth's gravity on the Moon, friction between roller skates and a rink floor, a banked roadway's force on a car, and forces on the tube of a spinning centrifuge.

• Frictional force(F_f): It is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other.

  • There are several types of friction: Dry friction, fluid friction, lubricated friction, etc...

Formula:

\({F_c} = \frac{{m{v^2}}}{r}\)

where F_c = centripetal force, m = mass, v = velocity, r = radius.

F_f = μ_s N

where μs = coefficient of friction,  N = normal force.

EXPLANATION:

As the car is moving on a circular road,

Force of friction provides the necessary centripetal force.

\({F_c} \le {μ _s}N = \frac{{m{v^2}}}{R}\)

\({v^2} \le \frac{{{μ _s}RN}}{m}\)

\({v^2} \le {μ _s}Rg..(N = mg)\)

v = √μ_sRg

Hence, the maximum speed of a car in a circular motion is v_max = √ μ_sRg

The correct option is 4.