Q

# Help me solve this A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere ?

A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere ?

• Option 1)

Rotational kinetic energy

• Option 2)

Moment of inertia

• Option 3)

Angular velocity

• Option 4)

Angular momentum

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As we have learned

Rotational angular momentum -

$\vec{L}= I\vec{w}$

- wherein

$I$ = Moment of inertia about fixed axis of rotation

$w$ = angular velocity of rotation

,

Law of conservation of angular moment -

$\vec{\tau }= \frac{\vec{dL}}{{dt}}$

- wherein

If net torque is zero

i.e. $\frac{\vec{dL}}{{dt}}= 0$

$\vec{L}= constant$

angular momentum is conserved only when external torque is zero .

when we have incraese radius of sphere keeping the mass same then moment of inertia increases

since there is no torque acting since it is rotating freely , angular momemntum remains constant

$L=Iw$

L = constant , I increases $\Rightarrow$ w decreases

Option 1)

Rotational kinetic energy

This is incorrect

Option 2)

Moment of inertia

This is incorrect

Option 3)

Angular velocity

This is incorrect

Option 4)

Angular momentum

This is correct

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