Question

Derive an expression for maximum speed of a vehicle moving along a horizontal circular track.

Hint:

The maximum speed of a vehicle in circular motion depends on the radius of the track, the coefficient of friction, and gravitational acceleration.

Answer 1

The maximum speed of a vehicle moving along a horizontal circular track occurs when the centripetal force is equal to the frictional force. The centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \] where \( m \) is the mass of the vehicle, \( v \) is the velocity, and \( r \) is the radius of the track. The frictional force \( F_f \) is given by:
\[ F_f = \mu mg \] where \( \mu \) is the coefficient of friction and \( g \) is the acceleration due to gravity. Equating the two forces:
\[ \frac{mv^2}{r} = \mu mg \] Solving for \( v \), we get the maximum speed:
\[ v_{{max}} = \sqrt{\mu g r} \]