# 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 $\inline \frac{R}{2}$ and   (iii) a solid sphere of radius $\inline \frac{R}{4}$ .  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)  $2:3:4$

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

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

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

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

Option 1)

$4:3:2$

Option 2)

$20:15:14$

Option 3)

$14:15:21$

Option 4)

$2:3:4$

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