Serway & Jewett — The Laws of Motion: Problem 45

A crate of weight \( F_g \) is pushed by a force \( \vec{P} \) on a horizontal floor as shown in Figure P5.45. The coefficient of static friction is \( \mu_s \), and \( \vec{P} \) is directed at angle \( \theta \) below the horizontal. (a) Show that the minimum value of \( P \) that will move the crate is given by \[ P = \frac{\mu_s F_g \sec \theta}{1 - \mu_s \tan \theta} \] (b) Find the condition on \( \theta \) in terms of \( \mu_s \) for which motion of the crate is impossible for any value of \( P \).