Question

A stone is tied to a string of length l and is whirled in a vertical circle with the other end of the string as the centre. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the change in velocity as it reaches a position where the string is horizontal (g being acceleration due to gravity) is?

Answer 1

let the direction of counter clockwise initial velocity

$=ui$

i is the directional of initial velocity

when the string is horizontal the body is covered the quarter of revolution the velocity is v j

j is the direction of velocity towards y axis

change in the velocity is $vj-ui$

magnitude of the change in velocity $\sqrt{v^2-u^2}$

by using the conservation of energy

energy of the body is $=1/2{mu}^2$

when it goes to quarter position KE spent is doing work mgh and remaing KE $=1/2{mv}^2$

$1/2m{u}^2=1/2{mv}^2+mgh$

$h=l$

$1/2m{u}^2=1/2{mv}^2+mgh$

$v^2-u^2=-2gl$

$v^2-u^2+2u^2={2u}^2-2gl$

$v^2-u^2+2u^2={2u}^2-gl$

$=\sqrt{v^2+u^2}=\sqrt{2}(u^2-gl)$