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An extensible string is wound over a rough pulley of mass M_(1) and radius R and a cylinder of mass M_(2) and radius R such that as the cylinder rolls down. The string unwounds over the pulley as well the cylinder. Findthe acceleration of cylinder M_(2).

Answer» <html><body><p></p>Solution :Through this illustration we will learn the application of torque equation and <a href="https://interviewquestions.tuteehub.com/tag/constraint-930945" style="font-weight:bold;" target="_blank" title="Click to know more about CONSTRAINT">CONSTRAINT</a> relation in more complex case here. We have change the block with cylinder `M_(2)` Pulley `M_(1)` will have oly pure rotation while cylinder `M_(2)` will have rotatioin and translation combined. Let us analyse step by step in the same way as the previous illustration. <br/> Step I: Analyse the motion of the pulley and the cyinder. Pulley: One <a href="https://interviewquestions.tuteehub.com/tag/rotational-625601" style="font-weight:bold;" target="_blank" title="Click to know more about ROTATIONAL">ROTATIONAL</a> acceleration `alpha_(1)` (clockwise) <br/> Cylinder: One rotational acceleration `alpha_(2)` (clockwise) andn a linear acceleration `a_(2)` (downward) <br/> Step II: Equation of motion for `M_(2)` <br/> `M_(2)g-T=M_(2)a_(2)` ..........i <br/> Step III: Torque equation for pulley `tau_(c)=I_(c)alpha` <br/> `TR=((M_(1)R^(2))/2)alpha_(1)implies(2T)/(M_(1)R)`.......ii <br/> Torquue equation for the cylinder (about centre of mass of cylinder `tau_(c)=I_(c)alpha` <br/> `implies TR=((M_(2)R^(2))/2)alpha_(2)` <br/> `alpha_(2)=(2T)/(M_(2)R)`.........iii <br/> Step IV: The acceleration of `P` and `Q` should be equal as both are connected with the same <a href="https://interviewquestions.tuteehub.com/tag/inextensible-2736845" style="font-weight:bold;" target="_blank" title="Click to know more about INEXTENSIBLE">INEXTENSIBLE</a> string ltbr. Accceleration of `P, a_(M)=alpha_(1)R` (downward) .........iv <br/> Acceleration of `Q, a_(N)=a_(2)-alpha_(2)R` (downwards) .............. v <br/> Hence, constraint relation `alpha_(1)R=a_(2)-alpha_(2)R` .............vi<br/> Step V: Solving equations <br/> After solving eqn i, ii, iii and iv, we <a href="https://interviewquestions.tuteehub.com/tag/get-11812" style="font-weight:bold;" target="_blank" title="Click to know more about GET">GET</a><br/> ` a_(2)=[((2M_(1)+M_(2)))/(3M_(1)+2M_(2))]g`</body></html>


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