Question

A body of mass $2\text{kg}$ is attached to a light rod of length $5\text{m}$ and is allowed to move in a vertical circle. The body is given an initial velocity of $u\text{m/s}$ at the bottom-most point, such that the final velocity at the top-most point is zero. The value of $u$ is
$($Take $g=10{\text{m/s}}^2)$

• A. $5\sqrt{2}m/s$
• B. $10m/s$
• C. $10\sqrt{2}m/s$
• D. $20\sqrt{2}m/s$

Answer 1

The correct option is C $10\sqrt{2}m/s$

Here at the top most point velocity $v=0$,

From the law of conservation of energy, let's say the bottom-most point is $A$ and the top-most point is $B$

$(K.E{)}_A+(U{)}_A=(K.E{)}_B+(U{)}_B$

Let potential at the bottom-most point be $U_A=0$
$\Rightarrow \frac{1}{2}m{u}^2+0=0+mg\left(2l\right)$

where $u$ is the initial velocity and $2l$ is the height of the mass from the bottom.
$\Rightarrow u=\sqrt{4gl}$
$u=\sqrt{4\times 10\times 5}$
$\Rightarrow u=\sqrt{200}$
$\Rightarrow u=10\sqrt{2}\text{m/s}$

Hence, option $\left(\text{C}\right)$ is the correct answer.