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

Consider the cylinder on inclined plane as shown in Fig . 7.73, then
(A) it will slide if mg sin `theta gt mu R`
i.e., mg sin `theta gt mu "mg cos" theta " " ["as" R = "mg cos" theta]`
i.e., tan `theta gt mu " " …… (a)`
(B) it will topple if
( mg sin `alpha) (h)/(2) gt "mg cos " alpha xx r " " ("replacing" theta by alpha)`
or tan `alpha gt (2 r //h) " " ....... (b) `
So , from Eqns . (a) and (b) , it is clear that :
(a) The cylinder will slide before toppling if
`alpha lt theta `
i.e., tan `theta lt "tan" alpha` or `mu lt (2 r//h)`
(b) The cylinder will topple before sliding if
`alpha lt theta `
i.e., tan `theta gt "tan" alpha ` or `mu gt (2r //h)`