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 R_{1} - µR_{2} = 0
R_{1} = µR_{2} ...(i)
∑V = 0 => R_{2} + µR_{1} = W ...(ii)
Putting value of R_{1} in equation (ii), we get
R_{2 } + µ2R_{2} = W
R_{2} = W/(1 + µ_{2}) ...(ii)
Putting value of R_{2} in equation (i), we get
R_{1} = µW/(1 + µ_{2}) ...(iv)
Taking the moment about O, we get
MO = µR_{1}r + µR_{2}r
= µr{R_{1} + R_{2}}
= µr{(µW/(1 + µ_{2})) + (W/(1 + µ_{2}))} = µrW{(1 + µ)/(1 + µ_{2})}
MO = µrW(1 + µ)/(1 + µ_{2}) .......ANS