Question

A body of mass m is suspended from a string of length l. What is the minimum horizontal velocity that should be given to the body at its lowest position so that it may complete one full revolution in the vertical plane with the point of suspension as the centre of the circle?

• A. $v=\sqrt{2lg}$
• B. $v=\sqrt{3lg}$
• C. $v=\sqrt{4lg}$
• D. $v=\sqrt{5lg}$

Answer 1

Answer: (D)

$v=\sqrt{5lg}$

$V_B$ is the minimum velocity given to the particle at the lowest point to complete the circle.The tendency of the string to become slack is maximum when the particle is at the topmost point of the circle.At the top, tension is given by $\Rightarrow T=\frac{m{V}_T^2}{l}-mg$

where $V_T=$ speed of the particle at the top. For $V_T$ to be minimum,

$T\approx 0\Rightarrow { V }_{ B }=\sqrt { 5gl } $

$ { V }_{ B } $be the critical velocity of the particle at the bottom, then from conservation of energy

$ \rightarrow mg(2l)+\dfrac { 1 }{ 2 } m{ { V }_{ T }^{ 2 } }=0+\dfrac { 1 }{ 2 } m{ { V }_{ B }^{ 2 } }$

putting $V_T=\sqrt{gl}$

Thus, $V_B=\sqrt{5gl}$