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A block of mass \mathrm{m} is attached to the pulley disc of equal mass \mathrm{m}  and radius \mathrm{r} by means of slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of \vartheta. Find its velocity when the string become taut.

Option: 1

\frac{2 \vartheta}{3}


Option: 2

\frac{ \vartheta}{3}


Option: 3

\frac{3 \vartheta}{2}


Option: 4

3\, \vartheta


Answers (1)

Using conservation of angular momentum about the hinge. If {\vartheta}' is the required velocity then

\mathrm{m\vartheta r =m \vartheta^{\prime} r+I_{c} \omega}
         \mathrm{=m \vartheta^{\prime} r+\frac{m r^{2}}{2} \cdot \frac{\vartheta^{\prime}}{r}}

\mathrm{m\vartheta r =m \vartheta^{\prime}\left[r+\frac{r^{2}}{2 r}\right]}
\mathrm{m \vartheta r =m \vartheta^{\prime}\left[\frac{2 r^{2}+r^{2}}{2 r}\right] \Rightarrow m \vartheta^{\prime}\left[\frac{3 r^{2}}{2 r}\right]}
\mathrm{m \vartheta r =m \vartheta^{\prime} \frac{3 r}{2} }
\mathrm{ \vartheta =\vartheta^{\prime} .\frac{3 }{2} }
\mathrm{ \vartheta^{\prime} = \frac{2\vartheta^{\prime} }{3} }  Ans.

Posted by

Ramraj Saini

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