Question

A small body is to be moved inside a vertical circular tube of radius 'l' . what minimum velocity should be imparted to it at its lowest point so that it can just complete the vertical circle?
a)√4gl
b)√5gl
c)√gl
d)√3gl​

Answer 1

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Answer 2

b) $\sqrt{5gl}$

Let the tension at point A of the loop be TA. We then have:  

Kinetic Energy at A = $\frac{1}{2} mv_{0}^2$                              (1)

Total Energy at point C = $\frac{1}{2}mv_{c}^2 + mg * 2l$       (2)

According to Newton's Second Law of Motion, at point C we get:

$T_{c} + mg = \frac{mv_{c}^2}{l}$

In order to complete the loop, $T_{c}$ >= 0

mg = $\frac{mv_{c}^2 }{l}$

∴ $v_{c}$ = $\sqrt[]{gl}$       (3)                                                        

From (1) & (2) using the principle of Conservation of Energy

= $\frac{1}{2}mv_{0}^2$ = $\frac{1}{2}mv_{c}^2 + 2mgl$

= $\frac{1}{2}mv_{0}^2$ = $\frac{1}{2}mgl + 2mgl$                (from (3) $v_{c}$ = $\sqrt[]{gR}$)

=  $v_{0}^2$ = gl+4gl

=  $v_{0}$ = $\sqrt[]{5gl}$