Question

A mass of m rests on a horizontal plane. The plane is gradually inclined until at an
angle theeta with the horizontal, the mass just begins to slide. Show that the coefficient
of static friction between the block and the surface is equal to tan theeta

Answer 1

see figure, here it is clearly shown that, a body of mass m placed on the surface of inclined plane.

component of weight along plane , W = mgsinθ

component of weight perpendicular to the plane = normal reaction acting on object by plane, R = mgcosθ

block has tendency to move downward due to weight of body along plane.

so, static frictional force acting upward direction to balance weight of body.

now at equilibrium,

weight of body along plane = Friction

or, mgsinθ = μR

or, mgsinθ = μmgcosθ

or, μ = tanθ

hence, it is clear that, coefficient of static friction is equal to tanθ