A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights $h_{s p h}$ and $h_{c y l}$ on the inline. The ratio $\frac{h_{s p h}}{h_{c y l}}$ is given by:

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TestHub · Accepted Answer

From conservation of mechanical energy,
For solid sphere: $\frac{1}{2} m_{s p h} V^{2} + \frac{1}{2} I_{s p h} ω^{2} = m_{s p h} g h_{s p h}$
$\frac{V^{2}}{2} + \frac{V^{2}}{5} = g h_{s p h}$
$\frac{7 V^{2}}{10} = g h_{s p h}$ …(i)
For cylinder: $\frac{1}{2} m_{c y l} V^{2} + \frac{1}{2} I_{c y l} ω^{2} = m_{c y l} g h_{c y l}$
$\frac{V^{2}}{2} + \frac{V^{2}}{4} = g h_{c y l}$
$\frac{3 V^{2}}{4} = g h_{c y l}$ …(ii)
From (i) and (ii) $\frac{h_{s p h}}{h_{c y l}} = \frac{14}{15}$

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