Q. 35

# When a body slides down from rest along a smooth inclined plane making an angle of 45° with the horizontal, it takes time T. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time pT, where p is some number greater than 1. Calculate the co-efficient of friction between the body and the rough plane.

Here, we have two cases, one with friction and one without friction. The distance travelled by the block in both cases is same.

Let the distance travelled by the block be “s” the mass of the block be “m”, the coefficient of friction be μ and the acceleration due to gravity be “g”. The angle made by the incline with the ground is θ .

Case:1 No friction

The equation of motion is

Here, initial velocity, u = 0

Acceleration in the incline frame of reference is, a=g(sinθ)

Case:2 With Friction

Here, we first calculate the acceleration.

From the free body diagram,

Using this acceleration in the equation of motion. The time taken is given by t’.

We know t’=pt

Substituting the values of t’, and t

Squaring both sides and simplifying,

Given, θ=45°

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