Question

QQ.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

Step-by-step explanation:

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

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.

Formula used :

$\: \:\:\: \:\:\:\: \:\:\:\:\: \:\:\: \:\: \:\:\: \:\:\:\: \:\:\:\:\underline{\boxed{\bf{\red{L=I\cdot\omega}}}}\green\bigstar$

• L denotes angular momentum

• I denotes moment of inertia

• ω denotes angular velocity

Calculation :

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

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

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

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

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

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

$:\implies \:\underline{\boxed{\frak{\pmb{\green{\omega=60\:rpm}}}}}\:\red{\bigstar}$