Figure shows a vertical force applied tangentially to a uniform cylinder of weight `F_(g)`. The coefficient of static friction between the cylinder and all surfaces is `0.500`. In terms of `F_(g)` , find the maximum force `P` that can be applied that does not cause the cylinder to rotate.

Explanation

When it is on the verge of slipping, the cylinder is in equilibrium.
`sumF_(x)=0 f_(1)=n_(2)mu_(s)n_(1)` and `f_(2)=mu_(s)n_(2)`
`sumF_(y)=0 P+n_(1)+f_(2)=F_(S)`
`sumtau=0 P=f_(1)+f_(2)`
As `P` grows so do `f_(1)` and `f_(2)`
therefore since `mu_(s)=1/2, f_(1)=(n_(1))/2` and `f_(2)=(n_(2))/2=(n_(1))/4`
then `P+n_(1)=(n_(1))/4=F_(g)`
`p=(n_(1))/2+(n_(1))/4=3/4n_(1)`
so `P+5/4n_(1)=F_(g)`
becomes `P=5/4(4/3)=F_(g)` or `8/3 P=F_(g)`

Therefore `P=3/8F_(g)`