friction problem

Question

To determine the coefficient of friction between a rough surface and a block, the surface is kept inclined at 45° and the block is released from rest. The block takes a time t in moving a distance d. The rough surface is then replaced by a smooth surface and the same experiment is repeated. The block now takes a time t/2 in moving down the same distance d. The coefficient of friction is

Answer 1

Normal force:

 

N = mg cosθ

 

Smooth surface:

 

d = a₁t₁²/2 = g sinθ (t/2)²/2 = g sinθ t²/8

 

Rough surface:

 

d = a₂t₂²/2 = (g sinθ - μN/m) t²/2 = (g sinθ - μg cosθ) t²/2

 

d = g(sinθ - μcosθ)t²/2

 

It is the same d for both the smooth and rough surfaces, so

 

g(sinθ - μcosθ)t²/2 = g sinθ t²/8

 

sinθ - μcosθ = sinθ /4

 

μ = (3/4)sinθ / cosθ = (3/4)tanθ = 0.75 tan45º = 0.75

 

μ = 0.75  [coefficient of kinetic friction]