A cylinder of mass M and radius r is mounted | vedclass

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(A) For the bucket of mass $m$,the equation of motion is:$mg - T = ma$  --- $(1)$For the cylinder of mass $M$ and radius $r$,the torque equation is:$\

Vedclass · Accepted Answer

$a = \frac{2mg}{M + 2m}$

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(A) For the bucket of mass $m$,the equation of motion is:$mg - T = ma$  --- $(1)$For the cylinder of mass $M$ and radius $r$,the torque equation is:$	au = I \alpha = rT$Since $I = \frac{1}{2}Mr^2$ and $\alpha = \frac{a}{r}$,we have:$\frac{1}{2}Mr^2 \left(\frac{a}{r}ight) = rT$$T = \frac{1}{2}Ma$  --- $(2)$Substituting equation $(2)$ into equation $(1)$:$mg - \frac{1}{2}Ma = ma$$mg = ma + \frac{1}{2}Ma$$mg = a \left(m + \frac{M}{2}ight)$$mg = a \left(\frac{2m + M}{2}ight)$$a = \frac{2mg}{M + 2m}$

$A$ cylinder of mass $M$ and radius $r$ is mounted on a frictionless axle over a well. $A$ rope of negligible mass is wrapped around the solid cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be

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