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 incline is same , the ratio of the maximum heights they climb is :

 

  • Option 1)

    4:3:2  

  • Option 2)

     20:15:14

  • Option 3)

     14:15:21

  • Option 4)

     2:3:4

 

Answers (1)

 

mgh=\frac{1}{m}\: V^{2}+\frac{1}{2}\: Iw^{2} =\frac{1}{2}\: mv^{2}+\frac{1}{2}I\: \frac{V^{2}}{R^{2}}

or           h=\frac{\left ( 1 +\frac{K^{2}}{R^{2}}\right )V^{2}}{2g}

\frac{K}{R}ring=1        \frac{K \; sphere}{R\; sphere}= \sqrt{\frac{2}{5}}      \frac{K \; sphere}{R\; sphere}= \sqrt{\frac{2}{5}}

=h_{1}:h_{2}:h_{3}\; =\; \; 20:15:14      


Option 1)

4:3:2  

Option 2)

 20:15:14

Option 3)

 14:15:21

Option 4)

 2:3:4

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