(A) For a solid cylinder of mass $m_2$ and radius $R$ undergoing downward motion,the equation of motion is $m_2g - T = m_2a_2$,where $T$ is the tensio

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(A) For a solid cylinder of mass $m_2$ and radius $R$ undergoing downward motion,the equation of motion is $m_2g - T = m_2a_2$,where $T$ is the tension in the string.For the rotation of the cylinder about its center of mass,the torque equation is $\tau = I\alpha_2$,where $I = \frac{1}{2}m_2R^2$ is the moment of inertia of the solid cylinder.The torque is provided by the tension $T$,so $TR = I\alpha_2 = (\frac{1}{2}m_2R^2)\alpha_2$.Thus,$T = \frac{1}{2}m_2R\alpha_2$.Since there is no slipping,the linear acceleration $a_2$ is related to the angular acceleration $\alpha_2$ by the condition $a_2 = R\alpha_2$.Therefore,$\alpha_2 = \frac{a_2}{R}$.

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

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In the given figure,a solid cylinder of mass $m_2$ and radius $R$ is suspended by a string wrapped around it,which passes over a fixed pulley of mass $m_1$ and radius $R$. If the linear acceleration of the solid cylinder is $a_2$ and there is no slipping,what is its angular acceleration $\alpha_2$?

A
$\frac{a_2}{R}$
B
$\frac{a_2 + g}{R}$
C
$\frac{2(a_2 + g)}{R}$
D
$\frac{2(g - a_2)}{R}$

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System of Particles and Rotational Motion

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Rotation Motion Basic, Motion of Connected Mass

Let $\vec{A}$ be a unit vector along the axis of rotation of a purely rotating body and $\vec{B}$ be a unit vector along the velocity of a particle $P$ of the body away from the axis. The value of $\vec{A} \cdot \vec{B}$ is

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The spool shown in the figure is placed on a rough horizontal surface and has an inner radius $r$ and an outer radius $R$. The angle $\theta$ between the applied force and the horizontal can be varied. The critical angle $\theta$ for which the spool does not roll and remains stationary is given by:

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$A$ thin rod is hinged at point $O$ and is in an unstable equilibrium position. It falls under the influence of gravity due to a slight disturbance. It makes angles of $60^{\circ}$,$90^{\circ}$,and $180^{\circ}$ with the vertical in positions $(2)$,$(3)$,and $(4)$ respectively. If $\omega_2$,$\omega_3$,and $\omega_4$ are the angular velocities in these positions,then:

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Four identical point masses $m$ are joined by light strings of length $l$ and arranged to form a square frame. The center of the table coincides with the center of the arrangement. If the arrangement rotates with a constant angular velocity $\omega$,find the tension in each string.

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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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In the given figure,a solid cylinder of mass $m_2$ and radius $R$ is suspended by a string wrapped around it,which passes over a fixed pulley of mass $m_1$ and radius $R$. If the linear acceleration of the solid cylinder is $a_2$ and there is no slipping,what is its angular acceleration $\alpha_2$?

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Let $\vec{A}$ be a unit vector along the axis of rotation of a purely rotating body and $\vec{B}$ be a unit vector along the velocity of a particle $P$ of the body away from the axis. The value of $\vec{A} \cdot \vec{B}$ is

The spool shown in the figure is placed on a rough horizontal surface and has an inner radius $r$ and an outer radius $R$. The angle $\theta$ between the applied force and the horizontal can be varied. The critical angle $\theta$ for which the spool does not roll and remains stationary is given by:

Four identical point masses $m$ are joined by light strings of length $l$ and arranged to form a square frame. The center of the table coincides with the center of the arrangement. If the arrangement rotates with a constant angular velocity $\omega$,find the tension in each string.

$A$ 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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