(C) For linear motion: $Mg - 2T = Ma$ For rotational motion: $2TR = I\alpha$ Here,$I = \frac{MR^2}{2}$ and $\alpha = \frac{a}{R}$. Substituting these

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(C) For linear motion: $Mg - 2T = Ma$ For rotational motion: $2TR = I\alpha$ Here,$I = \frac{MR^2}{2}$ and $\alpha = \frac{a}{R}$. Substituting these into the torque equation: $2TR = (\frac{MR^2}{2})(\frac{a}{R}) \implies 2TR = \frac{MRa}{2} \implies T = \frac{Ma}{4}$. Substituting $T$ into the linear motion equation: $Mg - 2(\frac{Ma}{4}) = Ma \implies Mg - \frac{Ma}{2} = Ma \implies Mg = \frac{3Ma}{2} \implies a = \frac{2}{3}g$. Now,find the tension $T$: $T = \frac{M}{4}(\frac{2}{3}g) = \frac{Mg}{6}$.

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

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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
$Mg, 4g$
B
$\frac{Mg}{2}, \frac{4}{3}g$
C
$\frac{Mg}{6}, \frac{2}{3}g$
D
$\frac{Mg}{3}, \frac{1}{3}g$

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System of Particles and Rotational Motion

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Rotation Motion Basic, Motion of Connected Mass

$A$ tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\omega$. The force exerted by the liquid on the tube at the other end is

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If $\vec{\omega} = (3\hat{i} + 4\hat{j} + 5\hat{k}) \text{ rad/s}$ and $\vec{r} = (\hat{i} - 2\hat{j} + 3\hat{k}) \text{ m}$,then the linear velocity vector of the particle is:

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$A$ homogeneous solid cylindrical roller of radius $R$ and mass $M$ is pulled on a cricket pitch by a horizontal force $F$ applied at its center. Assuming rolling without slipping,the angular acceleration of the cylinder is

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$A$ thin ring of radius $R$ is rotated about its center. Then its radius:

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$A$ straight rod of length $L$ is released on a frictionless horizontal floor in a vertical position. As it falls and slips,the distance of a point on the rod from the lower end,which follows a quarter circular locus,is

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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.

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$A$ tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\omega$. The force exerted by the liquid on the tube at the other end is

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