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Standard XII

Physics

Centripetal Acceleration

A uniform thi...

Question

A uniform thin rod $A B$ is equipped at both ends with the hooks shown and is supported by a frictionless horizontal table. Initially the rod is hooked at $A$ to a fixed pin $C$ about which it rotates with a constant angular velocity $ω_{1}$ . Suddenly end $B$ of the rod hits pin $D$ and gets hooked to pin $D$ , causing end $A$ to be released. Determine the magnitude of the angular velocity $ω_{2}$ of the rod in its subsequent rotation about $D$ . (Assume length and mass of the hook is negligible. Pin $C$ & $D$ are lying on a same horizontal line)

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Solution

Change in momentum about centre

$m v_{2} - m v_{1} = - N d t$

$m w_{2} \frac{L}{2} - m w_{1} \frac{L}{2} = - N d t$ ...........(i)

Chang e in angular momentum

$I_{1} w_{1} + I_{2} w_{2} = d t$

$I_{1} = I_{2}$

$\frac{m l^{2}}{12} (w_{1} + w_{2}) = N \times \frac{l}{2} d t$ ...........(ii)

from (1) and (2)

$\frac{m l}{6} (w_{1} + w_{2}) = \frac{m l}{2} (w_{1} - w_{2})$

$w_{1} + w_{2} = 3 w_{1} - 3 w_{2}$

$u w_{2} = 2 w_{1}$

$[w_{2} = \frac{w_{1}}{2}]$

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Change in momentum about centre mv2−mv1=−Ndtmw2L2−mw1L2=−Ndt...........(i)Chang e in angular momentumI1w1+I2w2=dtI1=I2ml212(w1+w2)=N×l2dt...........(ii)from (1) and (2)ml6(w1+w2)=ml2(w1−w2)w1+w2=3w1−3w2uw2=2w1[w2=w12]