The angle θ is slowly increased. write an expression for the angle at which the block begins to move in terms of μs.

Explanation:

An object is at rest along a slope if the net force acting on it is zero. The equation of the forces along the direction parallel to the slope is:

(1)

where

is the component of the weight parallel to the slope, with m being the mass of the object, g the acceleration of gravity, the angle of the slope

is the frictional force, with being the coefficient of friction and R the normal reaction of the incline

The equation of the forces along the direction perpendicular to the slope is

where

R is the normal reaction

is the component of the weight perpendicular to the slope

Solving for R,

And substituting into (1)

Re-arranging the equation,

This the condition at which the equilibrium holds: when the tangent of the angle becomes larger than the value of , the force of friction is no longer able to balance the component of the weight parallel to the slope, and so the object starts sliding down.

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