(C) According to the law of conservation of angular momentum,the external torque on the system is zero,so the angular momentum remains constant.$L_i =

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(C) According to the law of conservation of angular momentum,the external torque on the system is zero,so the angular momentum remains constant.$L_i = L_f$$I_1 \omega_1 = I_2 \omega_2$Here,$I_1 = \frac{1}{2} M R^2$ and $\omega_1 = \omega$.When the second disc is placed,the new moment of inertia $I_2$ is the sum of the moments of inertia of both discs:$I_2 = I_1 + I_{\text{disc2}} = \frac{1}{2} M R^2 + \frac{1}{2} (\frac{M}{4}) R^2 = \frac{1}{2} M R^2 (1 + \frac{1}{4}) = \frac{1}{2} M R^2 (\frac{5}{4}) = \frac{5}{8} M R^2$.Now,applying the conservation law:$(\frac{1}{2} M R^2) \omega = (\frac{5}{8} M R^2) \omega_2$$\omega_2 = \frac{1/2}{5/8} \omega = \frac{1}{2} \times \frac{8}{5} \omega = \frac{4}{5} \omega$.

Class 11 Physics System of Particles and Rotational Motion Conservation of angular momentum (combined translation and rotational motion)

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$A$ thin circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane with an angular velocity $\omega$ about an axis passing through its center and perpendicular to its plane. If another disc of the same radius but one-fourth of its mass is placed gently on top of it,what will be the angular velocity of the new system?

A
$\frac{5}{4}\,\omega$
B
$\frac{2}{3}\,\omega$
C
$\frac{4}{5}\,\omega$
D
$\frac{3}{2}\,\omega$

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System of Particles and Rotational Motion

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Conservation of angular momentum (combined translation and rotational motion)

$A$ uniform rod of length $8 a$ and mass $6 m$ lies on a smooth horizontal surface. Two point masses $m$ and $2 m$ moving in the same plane with speed $2 v$ and $v$ respectively strike the rod perpendicularly at distances $a$ and $2 a$ from the mid point of the rod in the opposite directions and stick to the rod. The angular velocity of the system immediately after the collision is:

DifficultTS EAMCET 2006

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$A$ metre stick is pivoted about its centre. $A$ piece of wax of mass $20 \,g$ travelling horizontally and perpendicular to it at $5 \,m/s$ strikes and adheres to one end of the stick so that the stick starts to rotate in a horizontal circle. Given the moment of inertia of the stick and wax about the pivot is $0.02 \,kg \cdot m^2$,the initial angular velocity of the stick is ........... $rad/s$.

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$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity:

MediumAIPMT 2009

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The position vector of a particle is given by $\vec{r} = 2\hat{i} - 6\hat{j} - 12\hat{k}$ units. $A$ force $\vec{F} = p\hat{i} + 3\hat{j} + 6\hat{k}$ units acts on it. For what value of $p$ is the angular momentum conserved?

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In the absence of torque,the rotational frequency of a body changes from $1 \text{ cy/sec}$ to $16 \text{ cy/sec}$. The ratio of the radius of gyration in the two cases will be: (in $: 1$)

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$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity:

The position vector of a particle is given by $\vec{r} = 2\hat{i} - 6\hat{j} - 12\hat{k}$ units. $A$ force $\vec{F} = p\hat{i} + 3\hat{j} + 6\hat{k}$ units acts on it. For what value of $p$ is the angular momentum conserved?

In the absence of torque,the rotational frequency of a body changes from $1 \text{ cy/sec}$ to $16 \text{ cy/sec}$. The ratio of the radius of gyration in the two cases will be: (in $: 1$)

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