A uniform cylinder of height `h` and radius `r` is placed with its circular face on a rough inclined plane and the inclination of the plane to the horizontal is gradually increased. If `mu` is the coefficient of friction, then under what condition the cylinder will (a) slide before toppling (b) topple before sliding.

Explanation

(A) Slide before toppling:
The block toppling :
mg `sin theta gt f `
`rArr mg sin theta gt mu mg cos theta `
`rArr tan theta gt mu `
i.e., block is at rest when
` tan theta le mu " '. . . (1)`
(B) Now suppose the friction f is sufficient to prevent sliding. Then we assume that N is shifted towards downward a distance x to prevent toppling Therefore. torque about O is zero.
`rArr f . (h)/(2) = N x`
`mg sin theta . (h)/(2) = mg cos theta . x `
` x = (tan theta h)/(2)`
Maximum value of x is a/2
So to prevent toppling ` x le (a)/(2)`
`rArr (tan theta . h)/(2) le a//2`
`rArr tan theta le (a)/(h) " ". . . (2)`
So, the block topple before sliding from (1) & (2) ` mu_(s) gt (a)/(h) `