Determine couple - Homogeneous cylinder:

Homogeneous cylinder having weight W rests on horizontal floor in contact with wall (Figure given below). If the coefficient of friction for the all contact surfaces be µ, determine couple M acting on cylinder, which will begin counter clockwise rotation.

Sol.: ∑H = 0 R₁  - µR₂ = 0

R₁ = µR₂                                                                                                                         ...(i)

∑V = 0  => R₂  + µR₁ = W                                                                                               ...(ii)

Putting value of R₁ in equation (ii), we get

R₂  + µ2R₂ = W

R₂ = W/(1 + µ₂)                                                                                                                   ...(ii)

Putting value of R₂ in equation (i), we get

R₁ = µW/(1 + µ₂)                                                                                                               ...(iv)

Taking the moment about O, we get

MO = µR₁r + µR₂r

= µr{R₁ + R₂}

= µr{(µW/(1 + µ₂)) + (W/(1 + µ₂))} = µrW{(1 + µ)/(1 + µ₂)}

MO = µrW(1 + µ)/(1 + µ₂)                                          .......ANS