Chapter 5, Problem 89AP

A crate of weight F_g is pushed by a force $\stackrel{\to }{P}$ on a horizontal floor as shown in Figure P5.45. The coefficient of static friction is μ_s, and $\stackrel{\to }{P}$ is directed at angle θ below the horizontal. (a) Show that the minimum value of P that will move the crate is given by

$P=\frac{{\mu }_s{F}_g\sec \theta }{1-{\mu }_s\tan \theta }$

Figure P5.45

(b) Find the condition on θ in terms of μ_s, for which motion of the crate is impossible for any value of P.