A thin uniform spherical shell and a uniform solid cylinder of the same mass and radius are allowed to roll down a fixed incline without slipping, starting from rest. The ratio tsphtcyl  of times taken by them to roll down the same distance is

From the free body diagram it is clear that we get the equation mgsinθ−f=ma.......1Torque  τ=fr=Iα⇒fr=I.ar.....2∵a=rα,pure rolling  Solving 1 and 2⇒mgsinθ−Iar2=ma      ∵I=mk2⇒mgsinθ−mk2ar2=ma⇒gsinθ=a1+k2r2⇒a=gsinθ1+k2r2Time taken by the bodies to roll down a distance ‘l’ ist=2la=2l1+k2R2gsinθFor shell :   k2r2=23For cylinder :  k2r2=12tsphtcy=1+231+12=5×23×3=103