$m g l \sin \theta=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}$ $=\frac{1}{2} m v^{2}+\frac{1}{2} \frac{m R^{2}}{2} \frac{v^{2}}{R^{2}}=\frac{3}{4

$\sqrt {3{m^2}{R^2}gl\,\sin \,\theta }$

$m g l \sin \theta=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}$ $=\frac{1}{2} m v^{2}+\frac{1}{2} \frac{m R^{2}}{2} \frac{v^{2}}{R^{2}}=\frac{3}{4} m v^{2}$ $\Rightarrow v=\sqrt{\frac{4 g l \sin \theta}{3}}$ $I=I_{c m} \omega+m v R=\frac{M R^{2}}{2} \frac{v}{R}+M v R=\frac{3 M v R}{2}$ $=\frac{3}{2} m R \sqrt{\frac{4 g l \sin \theta}{3}}=\sqrt{3 m^{2} R^{2} g l \sin \theta}$

English

6.System of Particles and Rotational MotionDifficult

A uniform disk of mass $m$ and radius $R$ rolls without slipping down an incline plane of length $l$ and inclination $\theta$. Initially the disk was at rest at the top of the incline plane. Its angular momentum about the point of contact with the inclined plane when it reaches the bottom will be equal to :-

A
$\sqrt {4{m^2}{R^2}gl\,\sin \,\theta }$
B
$\sqrt {3{m^2}{R^2}gl\,\sin \,\theta }$
C
$\sqrt {\frac{{{m^2}{R^2}gl\,\sin \,\theta }}{4}}$
D
zero

Std 11
Physics

A particle performs uniform circular motion with an angular momentum $L.$ If the angular frequency of the particle is doubled and kinetic energy is halved, its angular momentum becomes

Difficult

[AIEEE 2003]

View Solution

A rod of length $50\,cm$ is pivoted at one end. It is raised such that if makes an angle of $30^o$ fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad\,s^{-1}$ ) will be $(g = 10\,ms^{-2})$

Medium

[JEE MAIN 2019]

View Solution

$A$ uniform rod of mass $m$ and length $l$ hinged at its end is released from rest when it is in the horizontal position. The normal reaction at the hinge when the rod becomes vertical is :

Advanced

View Solution

A student of mass $M$ is $1.5 \,m$ tall and has her centre of mass $1 \,m$ above ground when standing straight. She wants to jump up vertically. To do so. she bends her knees so that her centre of mass is lowered by $0.2 \,m$ and then pushes the ground by a constant force F. As a result, she jumps up such that the maximum height of her feet is $0.3 \,m$ above ground. The ratio $F / Mg$ is

Advanced

[KVPY 2021]

View Solution

An air compressor is powered by a $200\,rad\,s^{-1}$ electric motor using a $V-$ belt drive. The motor pulley is $8\,cm$ in radius, and the tension in the $V-$ belt is $135\,N$ on one side and $45\,N$ on the other. The power of the motor will be ...... $kW$.

Medium

View Solution

Similar Questions

A particle performs uniform circular motion with an angular momentum $L.$ If the angular frequency of the particle is doubled and kinetic energy is halved, its angular momentum becomes

A rod of length $50\,cm$ is pivoted at one end. It is raised such that if makes an angle of $30^o$ fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad\,s^{-1}$ ) will be $(g = 10\,ms^{-2})$

$A$ uniform rod of mass $m$ and length $l$ hinged at its end is released from rest when it is in the horizontal position. The normal reaction at the hinge when the rod becomes vertical is :

An air compressor is powered by a $200\,rad\,s^{-1}$ electric motor using a $V-$ belt drive. The motor pulley is $8\,cm$ in radius, and the tension in the $V-$ belt is $135\,N$ on one side and $45\,N$ on the other. The power of the motor will be ...... $kW$.