A box of mass m rests on a rough, horizontal surface with a coefficient of static friction μs. If a force Fp arrow is applied to the box at an angle θ as shown, what is the minimum value of θ for which the box will not move regardless of the magnitude of Fp? (Use any variable or symbol stated above as necessary.) The arrow Fp point at the box from the top-left direction and the answer is not 90 degree.

Answer 1

So I assume the angle theta is above the horizontal.

downward force on surface=mg+Fp*sinTheta
friction force=mu( Fp*sinTheta+mg)
horizontal force=Fp*cosTheta
when it wont move, friction>horizontal
mu *Fp*sinTheta+mg*mu>Fp*costheta

Fp(mu*sinTheta-cosTheta)>=mg*mu
That is not easy to solve, however graphically it may be easier.