The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal place: \((i)\) a ring of radius \(R,\,(ii)\) a solid cylinder of radius \(\frac {R}{2}\)  nd \((iii)\) a solid sphere of radius \(\frac {R}{4}.\) If, in each case, the speed of the center of mass at bottom of the incline is same, the ratio of the maximum heights they climb is

If radius of gyration is, then

\(h = \frac{{\left( {1 + \frac{{{k^2}}}{{{R^2}}}} \right)}}{{2g}};\frac{{{k_{ring}}}}{{{R_{ring}}}} = 1,\frac{{{k_{solid\,cylinder}}}}{{{R_{solid\,cylinder}}}} = \frac{1}{{\sqrt 2 }}\)

\(\frac{{{k_{solid\,sphere}}}}{{{R_{solid\,sphere}}}} = \sqrt {\frac{2}{5}} \)

\({H_1}\,:\,{h_2}\,:\,{h_3}\,::\left( {1 + 1} \right)\,:\,\left( {1 + \frac{1}{2}} \right):\left( {1 + \frac{2}{5}} \right)::20\,:\,15\,:\,14\)

Therefore most appropriate option is \((B)\) 

Although which is not in correct sequence.