Question

A stone is tied to a string of length, $\ell$, and is whirled in a vertical circle with the other end of the string as the centre. At a certain instant of time, the stone is at its lowest position and has a speed, $u$. The magnitude of the change in velocity as it reaches a position where the string is horizontal ($g$ being acceleration due to gravity) is

• A. $\sqrt{2g\ell }$
• B. $\sqrt{2(u^2-g\ell )}$
• C. $\sqrt{u^2-g\ell }$
• D. $u-\sqrt{u^2-2g\ell }$

Answer 1

Answer: (B)

$\sqrt{2(u^2-g\ell )}$

$W_{mg}=\Delta K\Rightarrow -mg\ell =\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2$
or $m{v}^2=m(u^2-2g\ell )$ or $v=\sqrt{u^2-2g\ell }\hat{j}$
$\overline{u}=u\hat{i}$
$\therefore \overline{v}-\overline{u}=\sqrt{u^2-2g\ell }\hat{j}-u\hat{i}$
$\therefore |\overline{v}-\overline{u}|=\left[\right(u^2-2g\ell )+u^2{]}^{1/2}=\sqrt{2(u^2-g\ell )}$