A uniform sphere of mass `m` and radius `R` is placed on a rough horizontal surface. The spher is struck horizontally at a height h from the floor. Show that the sphere rolls without slipping with a constant velocity, when `h = 7R//5`.

Explanation

Consider the diagram where a sphere of m and radius ,R, struck horizontally at height h above the floor

the sphere will roll without sipping when `omega=(v)/( R)` , where v is linear velocity and `omega` is angular velocity of the sphere
Now , angular momentum of sphere , about centre of mass
( [We are applying conservation of angular momentum just momentum just before and after struck ]
`mv(h-R)=Iomega=((2)/(5)mR^(2))((v)/(R))`
` impliesmv(h-R)=(2)/(5)mvR`
`h-R=(2)/(5)Rimplies h=(7)/(5) R` Therefore the sphere eill roll without sipping with a constant velocity and hence , no loss of energy ,so `(D) to (i)`
Torque due to applied force ,F about centre of mass
For `tau=0,h=R` ,sphere will have only translational motion . it would lose energy by friction
hence , `(B ) to (iv)`
the sphere will spin clokwise when `taugt 0implies hgt R`
therefore , ` (C ) to (ii)`
the sphere will spin anti- clockwise when `tault 0implies hltR,(a) to (iii)`