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

Question

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

Infinity Learn · Accepted Answer

From the free body diagram it is clear that we get the equation mgsinθ−$$f=ma$$......$$.1Torque$$  τ$$=fr=I$$α⇒$$fr=I$$.ar....$$.2$$∵$$a=r$$α,pure rolling  Solving $$1$$ and $$2$$⇒mgsinθ−$$Iar2=ma$$      ∵$$I=mk2$$⇒mgsinθ−$$mk2ar2=ma$$⇒gsinθ$$=a1+k2r2$$⇒$$a=gsin$$θ$$1+k2r2Time$$ taken by the bodies to roll down a distance ‘l’ $$ist=2la=2l1+k2R2gsin$$θFor shell :   $$k2r2=23For$$ cylinder :  $$k2r2=12tsphtcy=1+231+12=5$$×$$23$$×$$3=103$$

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 $(\frac{t_{sph}}{t_{cyl}})$ of times taken by them to roll down the same distance is

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