Consider two heavy right circular rollers of the radii `R` and `r` respectively and rest on a rough horizontal plane a shown in figure. The larger roller has a string wound around it to which a horizontal force `P` can be applied as shown. Assuming that the coefficient of friction `m` has the same value for all surfaces of contact, determine the necessary condition under which the larger roller can be pulled over the smaller one. Assume the smaller cylinder should neither roll nor slide.

Options

A. `mu=sqrt((D)/(d))`
B. `mu=2sqrt((D)/(d))`
C. `mu=sqrt((d)/(D))`
D. `mu=2sqrt((D)/(d))`

Answer

C

Explanation

`sin alpha=(D-d)/(D+d)`
`cos alpha=(2sqrt(Dd))/(D+d)`

If smaller cylinder neither rolls nor slides then larger cylinder can be pulled over it. Then sum of moments about `C_(2)` should be zero
`muN_(1)d=muN_(2)d`
`N_(1)=N_(2)`
for no sliding
`muN_(2)+muN_(1) cos(90-alpha)=N_(1) cos alpha`
`implies mu (1+sin alpha)=cos alpha`