Question

A body is describing a vertical circle of radius r. What is the value of its minimum
speed at the bottom?​

Answer 1

To find:

Minimum speed for body at lowest point in vertical circle.

Calculation:

First , we need to find the minimum velocity at highest point :

$\therefore \: mg + T = \dfrac{m {u}^{2} }{r}$

$\implies \: mg + 0 = \dfrac{m {u}^{2} }{r}$

$\implies \: {u}^{2} = gr$

$\implies \: u = \sqrt{gr}$

Now, applying CONSERVATION OF MECHANICAL ENERGY upto lowest point:

$\therefore \: KE1 + PE1 = KE2 + PE2$

$\implies \: \dfrac{1}{2} m {v}^{2} + 0 = \dfrac{1}{2} m {u}^{2} + mg(2r)$

$\implies \: \dfrac{1}{2} m {v}^{2} + 0 = \dfrac{1}{2} m {( \sqrt{gr} )}^{2} + mg(2r)$

$\implies \: \dfrac{1}{2} m {v}^{2} + 0 = \dfrac{1}{2} m (gr) + mg(2r)$

$\implies \: \dfrac{1}{2} {v}^{2} + 0 = \dfrac{1}{2} (gr) + g(2r)$

$\implies \: {v}^{2} + 0 = (gr) + 4gr$

$\implies \: {v}^{2} = 5gr$

$\implies \: v = \sqrt{5gr}$

So, minimum velocity at lowest point is √(5gr).