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Obtain an expression for the speed with which a vehicle can safely negotiate a flat  curved road.

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Consider a car of mass 'm' moving with a constant speed 'v' on a flat horizontal road of radius 'r'.

The various forces acting on the car are:

(i) The weight of the car acting vertically downward, i.e.,

W = mg     ....(i)

(ii) The normal reaction (R) of the ground on the car which is equal and opposite to the weight of the car, i.e., R = mg.

Since the car is moving on a circular path, so it requires centripetal force,

Fc = mv2/r

Then centripetal force must be provided by the frictional force (F) between the tyres of the car and the ground.

Force of friction F = μR = μ mg    ....(ii)

The car remains moving on the road, if

mv2/r = μ mg ⇒ v = √{μ gr}    ...(iii)

The car should negotiate the flat circular path not more than that given by equ. (iii). If the speed of the car becomes more than the speed given by eqn. (i) then the centripetal force needed by the car will not be provided by the force of friction and hence the car will skid and go off the road.

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