Search any question & find its solution
Question:
Answered & Verified by Expert
A uniform solid spherical ball is rolling down a smooth inclined plane from a height $h$. The velocity attained by the ball when it reaches the bottom of the inclined plane is v. If the ball is now thrown vertically upwards with the same velocity $v$, the maximum height to which the ball will rise is
Options:
Solution:
1556 Upvotes
Verified Answer
The correct answer is:
$\frac{5 h}{7}$
We know that total kinetic energy of a body rolling without slipping
$K_{\text {total }}=K_{\text {rot }}+K_{\text {trans }}$ For solid spherical ball, $I=\frac{2}{5} m R^{2}$ (along to diameter) and $v=R \omega$ where $R$ is radius of spherical ball
So.
$$
K_{\text {total }}=\frac{1}{2}\left(\frac{2}{5} m R^{2}\right) \omega^{2}+\frac{1}{2} m R^{2} \omega^{2}
$$
$$
\begin{array}{l}
=\frac{7}{10} m R^{2} \omega^{2} \\
K=\frac{7}{10} m v^{2}
\end{array}
$$
Potential energy = Kinetic energy
$$
\begin{aligned}
m g h &=\frac{7}{10} m v^{2} \\
v^{2} &=\frac{10}{7} g h
\end{aligned}
$$
For vertical projection,
$$
\begin{array}{c}
v^{2}=u^{2}+2 g h^{\prime} \\
\frac{10}{7} g h=0+2 g h^{\prime} \Rightarrow h^{\prime}=\frac{5}{7} h
\end{array}
$$
$K_{\text {total }}=K_{\text {rot }}+K_{\text {trans }}$ For solid spherical ball, $I=\frac{2}{5} m R^{2}$ (along to diameter) and $v=R \omega$ where $R$ is radius of spherical ball
So.
$$
K_{\text {total }}=\frac{1}{2}\left(\frac{2}{5} m R^{2}\right) \omega^{2}+\frac{1}{2} m R^{2} \omega^{2}
$$
$$
\begin{array}{l}
=\frac{7}{10} m R^{2} \omega^{2} \\
K=\frac{7}{10} m v^{2}
\end{array}
$$
Potential energy = Kinetic energy
$$
\begin{aligned}
m g h &=\frac{7}{10} m v^{2} \\
v^{2} &=\frac{10}{7} g h
\end{aligned}
$$
For vertical projection,
$$
\begin{array}{c}
v^{2}=u^{2}+2 g h^{\prime} \\
\frac{10}{7} g h=0+2 g h^{\prime} \Rightarrow h^{\prime}=\frac{5}{7} h
\end{array}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.