Free Body Diagram

Impending Motion

In this problem, the ramp AB may slide if the friction force at A is too low. To find the coefficient of friction that will prevent the ramp from sliding, the conditions under which the ramp will be in motion needs to be determined.

First, draw a free-body diagram of the ramp with the friction acting to the left since the ramp will want to slide to the right.

Using equilibrium equations, the moment about point A can be found as,

     ΣM_A = N_B (3 m) - F_C (2 m) = 0

     N_B = (1,000 N) (2 m) / (3 m) = 666.7 N

Since motion is impending at each point and the normal force at point B is known, the friction at point B is

     f_B = μ_B N_B = (0.2)(666.7 N) = 133.3 N

The equilibrium equations can now be used to find the normal and frictional forces at point A.

     ΣF_x = 0

     f_A = F_C sinα - N_B sinα - f_B cosα
         = (1,000 - 666.7) sin25 - 133.3 cos25
         = 20.0 N

     ΣF_y = 0

     N_A = F_C cosα - N_B cosα - f_B sinα
         = (1,000 - 666.7) cos25 - 133.3 sin25
         = 358.4 N

Now that the normal and frictional forces are known, the coefficient of friction can be determined as

     μ_A = f_A/N_A = 0.056

This is the lowest value of μ_A needed to prevent motion. For lower values of μ_A, the ramp is in motion, and for higher values, the ramp is in equilibrium.