Example 5: If the block weighs 100 lb and μ_s = 0.3, determine the force P needed for impending sliding. Use

a. θ = 30°
b. θ = 0°
c. θ = -30°
d. Find P_min and the corresponding angle θ.

Solution:

We begin by drawing the free-body diagram of the block and writing the two force equilibrium equations.

Since the condition is impending slipping, the friction force is related to the normal force as

Substituting for F and eliminating N gives

For the three specified angles, we get

a. θ = 30°, P= 41.9 lb.
b. θ = 0°, P= 30.0 lb.
c. θ = -30°, P= 29.5 lb.

To find the minimum P, we need to first find the corresponding θ as

Substituting this value into the function P gives