(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:$\

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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:$\tau = I \alpha = rT$Since $I = \frac{1}{2}Mr^2$ and $\alpha = \frac{a}{r}$,we have:$\frac{1}{2}Mr^2 \left(\frac{a}{r}\right) = 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$$mg = a \left(m + \frac{M}{2}\right)$$mg = a \left(\frac{2m + M}{2}\right)$$a = \frac{2mg}{M + 2m}$

Class 11 Physics System of Particles and Rotational Motion Rotation Motion Basic, Motion of Connected Mass

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$A$ 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

A
$a = \frac{2mg}{M + 2m}$
B
$a = \frac{mg}{M + m}$
C
$a = \frac{mg}{M + 2m}$
D
$a = \frac{Mg}{M + 2m}$

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System of Particles and Rotational Motion

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Rotation Motion Basic, Motion of Connected Mass

$A$ solid cylinder of mass $M$ and radius $R$ is placed in a horizontal position. Two strings are wrapped around the cylinder. Find the tension in the strings and the acceleration of the cylinder as the strings unwind.

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What is a rigid body? Explain the difference between a rigid body and a solid body.

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$A$ 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 cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be

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$A$ rigid lamina is rotating about an axis passing perpendicular to its plane through point $O$ as shown in the figure. The angular velocity of point $B$ with respect to $A$ is ....... $rad/s$.

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$A$ 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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$A$ solid cylinder of mass $M$ and radius $R$ is placed in a horizontal position. Two strings are wrapped around the cylinder. Find the tension in the strings and the acceleration of the cylinder as the strings unwind.

$A$ disc at rest is subjected to a uniform angular acceleration about its axis. Let $\theta$ and $\theta^{\prime}$ be the angle made by the disc in the $2^{\text{nd}}$ and $3^{\text{rd}}$ second of its motion,respectively. The ratio $\frac{\theta}{\theta^{\prime}}$ is

What is a rigid body? Explain the difference between a rigid body and a solid body.

$A$ 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 cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be

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