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. Then the minimum value of $h$ for a successful looping is (neglect friction).

• A. $R$
• B. $\frac{R}{2}$
• C. $\frac{R}{4}$
• D. $2R$

Answer 1

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

As we know, $P.E.+K.E=Const.$

At lowest point in circle car will have maximum kinetic energy and at top point in circle it will have some K.E .This K.E. is because of decrease in P.E. from initial point i.e $mgh$


From free body diagram at top point (B) of circle,
$mg+N=\frac{m{v}^2}{R}$
$N=\frac{m{v}^2}{R}-mg$
at this point B, $N=0$ for minimum height.
$\therefore \frac{m{v}^2}{R}=mg$
$v=\sqrt{Rg}$
As decrese in potential energy is equal to gain in kinetic energy by equating them
$\frac{1}{2}m{v}^2=mg{h}_{min}$
$\frac{1}{2}m\left(Rg\right)=mg{h}_{min.}$
$h_{min.}=\frac{R}{2}$