(D) The angular momentum $L$ of the stone is given by $L = mvr = mr^2\omega$,where $m$ is mass,$v$ is linear velocity,$r$ is the radius,and $\omega$ i

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(D) The angular momentum $L$ of the stone is given by $L = mvr = mr^2\omega$,where $m$ is mass,$v$ is linear velocity,$r$ is the radius,and $\omega$ is angular velocity.Since the angular momentum is constant,we have $mr^2\omega = C$ (where $C$ is a constant).This implies $\omega = \frac{C}{mr^2}$.The tension $T$ in the string provides the necessary centripetal force for circular motion:$T = m\omega^2r$.Substituting the expression for $\omega$ into the tension formula:$T = m \left( \frac{C}{mr^2} \right)^2 r = m \left( \frac{C^2}{m^2r^4} \right) r = \frac{C^2}{mr^3} = \left( \frac{C^2}{m} \right) r^{-3}$.Comparing this with $T = Ar^n$,we identify $A = \frac{C^2}{m}$ and $n = -3$.

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

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$A$ stone of mass $m$,tied to the end of a string,is whirled around in a circle on a horizontal frictionless table. The length of the string is reduced gradually,keeping the angular momentum of the stone about the centre of the circle constant. Then,the tension in the string is given by $T = Ar^n$,where $A$ is a constant and $r$ is the instantaneous radius of the circle. The value of $n$ is equal to

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$-1$
B
$-2$
C
$-4$
D
$-3$

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System of Particles and Rotational Motion

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Conservation of angular momentum (combined translation and rotational motion)

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EasyAIIMS 2004

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$A$ thin circular ring of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity $\omega$. If two objects each of mass $m$ are attached gently to the opposite ends of a diameter of the ring,the ring will then rotate with an angular velocity:

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$A$ boy runs and sits on a rotating platform. Which of the following quantities will be conserved?

$A$ rigid body of mass $2 \ kg$ is moving in a circular path of radius $0.8 \ m$ with an angular velocity of $44 \ rad \ s^{-1}$. If the radius of the circular path becomes $1 \ m$,the new angular velocity of the body will be ........ $rad \ s^{-1}$.

$Assertion$ : There are very small sporadic changes in the speed of rotation of the earth.$Reason$ : Shifting of large air masses in the earth's atmosphere produces a change in the moment of inertia of the earth, causing its speed of rotation to change.

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$Assertion$ : There are very small sporadic changes in the speed of rotation of the earth.
$Reason$ : Shifting of large air masses in the earth's atmosphere produces a change in the moment of inertia of the earth, causing its speed of rotation to change.