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  4. A solid sphere rolls without slipping, first horizontally and then up to a point X at height h on an inclined plane before rolling down, as shown in the figure below. <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/physics/a114857c980508729c0007c1703385fc-.jpeg /> The initial horizontal speed of the sphere is

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Q. A solid sphere rolls without slipping, first horizontally and then up to a point $X$ at height h on an inclined plane before rolling down, as shown in the figure below.
image
The initial horizontal speed of the sphere is

KVPYKVPY 2013Work, Energy and Power

Solution:

Let initial horizontal speed of sphere is $\upsilon$. Then, total kinetic energy of sphere on horizontal part
$=\left(KE\right)_{\text{translation}} +\left(KE\right)_{\text{rotation}}$
$=\frac{1}{2}m\upsilon^{2} +\frac{1}{2}I\omega^{2} $
$=\frac{1}{2}m\upsilon^{2}+\frac{1}{2}\times\frac{2}{5}mR^{2}\times\frac{\upsilon^{2}}{R^{2}}=\frac{7}{10}m\upsilon^{2} $
If sphere rises upto height $h$ then, by conservation of energy, we have
mgh $=\frac{7}{10}m\upsilon^{2} $
or $\upsilon=\sqrt{\frac{10}{7}gh }$