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:

Question

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:

Aicharya · 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}

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see…

Solution