A block with a mass of 4.25 kg is placed at rest on a surface inclined at an angle of 44.8° above the horizontal. The coefficient of static friction between the block and the surface is 0.520, and a force of magnitude $F$ pushes upward on the block, parallel to the inclined surface. The block will remain at rest only if $F$ is greater than a minimum value, Fmin, and less than a maximum value, Fmax. Calculate Fmin.

Question

A block with a mass of 4.25 kg is placed at rest on a surface inclined at an angle of 44.8° above the horizontal. The coefficient of static friction between the block and the surface is 0.520, and a force of magnitude $F$ pushes upward on the block, parallel to the inclined surface. The block will remain at rest only if $F$ is greater than a minimum value, Fmin, and less than a maximum value, Fmax. Calculate Fmin.

Answer 1

Fmax occurs when the block moves.

frictionforce=mu(mg)cosTheta
weight down plane=mgSinTheta

Net force up the plane=Fmin-mg(sinTheta-mu*CosTheta0.

so if the net force ukp the plane is zero, then it remains still, or
Fmin=mg(sinTheta-muCosTheta)

Now, the max value occurs when Fmax=weightdown+friction or
Fmax=mg(sinTheta+mu*CosTheta)