• # question_answer 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 centre of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is : [JEE Main 9-4-2019 Morning] A) 4 : 3 : 2                       B) 14 : 15 : 20C) 10 : 15 : 7D) 2 : 3 : 4

$\frac{1}{2}\left( m+\frac{I}{{{R}^{2}}} \right){{\text{v}}^{2}}=mgh$ if radius of gyration is k, then $h=\frac{\left( 1+\frac{{{k}^{2}}}{{{R}^{2}}} \right){{\text{v}}^{2}}}{2g},\frac{{{k}_{ring}}}{{{R}_{ring}}}=1,\frac{{{k}_{solid\,cylinder}}}{{{R}_{solid\,cylinder}}}=\frac{1}{\sqrt{2}}$ $\frac{{{k}_{solid\,sphere}}}{{{R}_{solid\,sphere}}}=\sqrt{\frac{2}{5}}$ ${{h}_{1}}:{{h}_{2}}:{{h}_{3}}::(1+1):\left( 1+\frac{1}{2} \right):\left( 1+\frac{2}{3} \right)::20:15:14$ Therefor most appropriate option is [b] athough which in not in correct sequence