Evaluate the angle - block impends down the plane:

What should be value of the angle shown in the figure so that the motion of the 90N block impends down the plane? The coefficient of friction for entire surface = 1/3.

Sol.: Consider equilibrium of block 30N             

∑V = 0;                                                                      

R₁  - 30cos θ  = 0,

R₁ = 30cos                                 ...(i)

∑H = 0;

T - µR₁ - 30sinθ   = 0,

T = 10cos θ  + 30sinθ           ...(ii)

Consider the equilibrium of block 90N

∑V = 0;

R₂  - R₁  - 90cos θ  = 0

R₂ = 120cos                             ...(iii)

∑H = 0;

90sin θ  - µR₁ - µR₂ = 0,

90sin θ  = 10cos θ  + 40sinθ         ...(vi)

tanθ = 5/9 that is  θ = 29.050