The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: (i ) a ring of radius R , ( ii ) a solid cylinder of radius R2 and ( iii ) a solid sphere…

Question

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane:

(i)

a ring of radius

R, (ii)

a solid cylinder of radius

\frac{R}{2}

and

(iii)

a solid sphere of radius

\frac{R}{4}

. If, in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is :

Accepted Answer

Correct answer: B. (b) mgh= 12mv_ cm^2+ 12I_ cm ^2 = 12mv_ cm^2+ 12I_ cm ( v_ cmR ) = 12 (m+ I_ cmR^2 )v_ cm^2= 1m2 [1+ K^2R^2 ] ∴h ∝1+ K^2R^2For ring :h ∝2 ( ∵K=R ) For solid cylisnder, h ∝ 32 ( ∵K= R 2 ) For solid sphere, h ∝ 75 ( ∵K= 25R ) Ratio of heights 2 : 32 : 75 ⇒20 :15 :14

Solution

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