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A T-shaped system is made by using two identical rods AB and CD each of mass m and length ‘l’. A linear impulse J is imparted to the system at midpoint of rod AB. Just after impulse, angular speed of rod AB is (1) $ω_{1}$ if midpoint of rod CD is connected to end B by a pin support so that rod CD can rotate freely about hinge at end B and (2) $ω_{2}$ if midpoint of CD is rigidly connected to end B. Find $\frac{4 ω_{1}}{ω_{2}}$

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answer is 4.25.

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Detailed Solution

$J \frac{l}{2} = (\frac{m l^{2}}{3} + m l^{2}) ω_{1}$

$J \frac{l}{2} = (\frac{m l^{2}}{3} + (\frac{m l^{2}}{12} + m l^{2})) ω_{2}$

$∴ \frac{4 ω_{1}}{ω_{2}} = \frac{17}{4} = 4.25$

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