Question

A small ball is rolled with speed $u$ from point $A$ along a smooth circular track as shown in figure. If $x=3R$, then determine the point of projection after passing through $C$, the highest point.

• A. $\frac{3}{2}\sqrt{gR}$
• B. $\frac{1}{2}\sqrt{gR}$
• C. $\frac{5}{3}\sqrt{gR}$
• D. $\frac{5}{2}\sqrt{gR}$

Answer 1

The correct option is D $\frac{5}{2}\sqrt{gR}$


From C the ball becomes a horizontal projectile with range $x=3R$ and height $H=2R$. Thus,
$x=v\sqrt{\frac{2\left(H\right)}{g}}$
$3R=v\sqrt{\frac{2\left(2R\right)}{g}}$
$\Rightarrow v=\frac{3}{2}\sqrt{gR}$
Conserving mechanical energy from $B$ to $C$.
$K_B+U_B=K_C+U_C$
$\frac{1}{2}m{u}^2+0=\frac{1}{2}m{v}^2+mg\left(2R\right)$
$u^2=\frac{9}{4}gR+4gR\Rightarrow u=\frac{5}{2}\sqrt{gR}$