(C) For the falling mass $m$,the equation of motion is: $mg - T = ma$ $(1)$For the rotating cylinder,the torque equation is: $\tau = I\alpha = I(a/r)

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${\left( {\frac{{2mgh{r^2}}}{{I + m{r^2}}}} \right)^{1/2}}$

(C) For the falling mass $m$,the equation of motion is: $mg - T = ma$ $(1)$For the rotating cylinder,the torque equation is: $\tau = I\alpha = I(a/r) = T \cdot r$,which gives $T = \frac{Ia}{r^2}$ $(2)$Substituting $(2)$ into $(1)$: $mg - \frac{Ia}{r^2} = ma$$mg = a(m + \frac{I}{r^2}) = a(\frac{mr^2 + I}{r^2})$Thus,the acceleration $a = \frac{mgr^2}{I + mr^2}$Using the kinematic equation $v^2 = u^2 + 2ah$ with $u = 0$:$v^2 = 2 \cdot (\frac{mgr^2}{I + mr^2}) \cdot h$$v = \sqrt{\frac{2mghr^2}{I + mr^2}}$

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

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$A$ string is wrapped around a cylinder of radius $r$ and moment of inertia $I$. $A$ mass $m$ is attached to one end of the string. The cylinder can rotate freely about its horizontal axis. If the mass $m$ is released from rest at a height $h$,what will be its velocity?

A
$(2gh)^{1/2}$
B
${\left( {\frac{{2mgh{r^2}}}{I}} \right)^{1/2}}$
C
${\left( {\frac{{2mgh{r^2}}}{{I + m{r^2}}}} \right)^{1/2}}$
D
${\left( {\frac{{mgh{r^2}}}{{I + 2m{r^2}}}} \right)^{1/2}}$

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System of Particles and Rotational Motion

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Rotation Motion Basic, Motion of Connected Mass

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$A$ string is wrapped around a cylinder of radius $r$ and moment of inertia $I$. $A$ mass $m$ is attached to one end of the string. The cylinder can rotate freely about its horizontal axis. If the mass $m$ is released from rest at a height $h$,what will be its velocity?

Similar Questions

What is the value of linear velocity if $\overrightarrow{r} = 3\widehat{i} + 4\widehat{j} + 6\widehat{k}$ and $\overrightarrow{\omega} = -5\widehat{i} + 3\widehat{j} + 5\widehat{k}$?

In the following figure,a body of mass $m$ is tied at one end of a light string and this string is wrapped around a solid cylinder of mass $M$ and radius $R$. At the moment $t = 0$,the system starts moving. If the friction is negligible,the angular velocity at time $t$ would be

$A$ 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

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