Question

Figure shows a loop-the-loop track of radius R. A car (without engine) starts 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. Neglect friction

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

Answer 1

The correct option is A $\frac{R}{2}$

Suppose the speed of the car at the topmost point of the loop is v. Taking the gravitational potential energy to be zero at the platform and assuming that the car starts with a negligible speed, the conservation of energy shows,
$0=-mgh+\frac{1}{2}m{v}^2$
or $m{v}^2=2mgh$ .......(i)
where m is the mass of the car. The car moving in a circle must have radial acceleration $\frac{v^2}{R}$ at this instant. The forces on the car are, mg due to gravity and N due to the contact with the track. Both these forces are in radial direction at the top of the loop. Thus, from Newton’s Law,
$mg+N=\frac{m{v}^2}{R}$
or, $mg+N=\frac{2mgh}{R}$.
For h to be minimum, N should assume the minimum value which can be zero. Thus,
$2mg\frac{h_{min}}{R}=mg\Rightarrow h_{min}=\frac{R}{2}$.