Question

Figure shows a loop track of radius $r$. A box starts sliding from a platform at a distance ($h$) above the top of the loop and goes around the loop without falling off the track. Find the minimum value of $h$ for a successful looping. Friction is negligible on all surfaces.

• A. $2r$
• B. $r$
• C. $\frac{r}{2}$
• D. $\frac{3r}{2}$

Answer 1

The correct option is C $\frac{r}{2}$

Let the box be of mass ($m$)
Let point $B$ be taken as a reference point,
Applying law of conservation of energy between $A$ and $B$
$\Rightarrow mg(h+2r)=\frac{1}{2}m{v}^2$
$v=\sqrt{2g(h+2r)}$
To complete the loop, the box must have a speed of at least $\sqrt{5gr}$ at the bottom, otherwise it will loose contact and start following the projectile parabolic trajectory. Thus we have
$\sqrt{2g(h+2r)}\ge \sqrt{5gr}$
$h+2r\ge \frac{5}{2}r$
$\Rightarrow h\ge \frac{r}{2}$