A bob of mass m , suspended by a string of length l 1 , is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another

Question

A bob of mass m , suspended by a string of length l 1 , is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m , suspended by a string of length l 2 , which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio l 1 l 2 is

TestHub · Accepted Answer

As shown in the figure shown above, the minimum speed required by the bob of mass $m$ attached to the string of length $l_{1}$ in order to complete the full circle is $\sqrt{5 g l_{1}}$ .

When the bob goes to the highest point, its speed becomes $\sqrt{g l_{1}}$ . Here, it strikes with the identical mass, which is tied to the string of length $l_{2}$ . The speed delivered to the object is $\sqrt{g l_{1}}$ . This should be equal to $\sqrt{5 g l_{2}}$ so that this bob could complete a full circle.
$\sqrt{g l_{1}} = \sqrt{5 g l_{2}}$
$\Rightarrow g l_{1} = 5 g l_{2}$
$\Rightarrow \frac{l_{1}}{l_{2}} = 5$

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