Question

Q] Two wheels of moment of inertia 4 kgm² rotate side by side at the rate of 120 rev/min and 240 rev/min respectively in the opposite directions. If now both the wheels are coupled by means of a weightless shaft so that both the wheels rotate with a common angular speed.Calculate the new speed of rotation​

Answer 1

Given :

Two wheels of moment of inertia 4 kgm² rotate side by side at the rate of 120 rpm and 240 rpm respectively in the opposite directions.

Both wheels are brought in contact so that they rotate with a common angular speed.

To Find :

New speed of rotation.

Concept :

Since no external torque acts on the whole system throughout the coupling process; angular momentum of the whole system is conserved.

• Angular momentum is measured as the product of moment of inertia and angular velocity.
• It is a scalar quantity having only magnitude.

Formula : $\underline{\boxed{\bf{\gray{L=I\cdot\omega}}}}$

• L denotes angular momentum

• I denotes moment of inertia

• ω denotes angular velocity

Calculation :

$\sf:\implies\:I\omega_1-I\omega_2=(2I)(\omega)$

• where ω denotes angular velocity of the combination and negative sign shows opposite direction of rotation.

$\sf:\implies\:\omega_1-\omega_2=2\omega$

$\sf:\implies\:240-120=2\omega$

$\sf:\implies\:120=2\omega$

$\sf:\implies\:\omega=\dfrac{120}{2}$

$:\implies\:\underline{\boxed{\bf{\orange{\omega=60\:rpm}}}}$

Answer 2

${ \underline{ \underline{ \huge{ \rm{ \red{SOLUTION}}}}}}$

${ \dag{ \underline{ \underline{ \tt{ \orange{ \: \: to \: find}}}}}} \\ \\ { \sf{ \:new \: speed \: of \: rotation...}}$

${ \dag{ \underline{ \underline{ \tt{ \orange{ \: \: formula \: used}}}}}} \\ \\ { \boxed{ \boxed{ \sf{ \green{angular \: momentum = I \: \times \: ω}}}}} \\ \\ { \sf{ \: where}} \\ { \sf{ \: l = { \green{moment \: of \: inertia \: }}}} \\ { \sf{ \: ω = { \green{angular \: speed}}}}$

${ \dag{ \underline{ \underline{ \orange{ \tt{ \: \: concept \: used}}}}}} \\ \\ { \boxed{ \boxed{ \green{ \sf{conservation \: of \: angular \: momentum}}}}}$

The law of conservation of angular momentum states that when no external torque acts on an object, no change of angular momentum will occur.

${ \underline{ \sf{ \red{initial \: angular \: momentum}}}} { \sf{ \: = lω2 - lω1}}$

${ \sf{ \underline{ \red{final \: angular \: momentum}}}}{ \sf{ = \: l ω \: + l ω}}$

${ \boxed{ \boxed{ \green{ \sf{ω = 60 \: rev \: {min}^{ - 1}}}}}}$