Two discs of moments of inertia I1 and I2 about their...

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes, rotating with angular frequencies $\omega_{1}$ and $\omega_{2}$ respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be (a) $\frac{I_{1} \omega_{1}+I_{2} \omega_{2}}{I_{1}+I_{2}}$ (b) $\frac{I_{2} \omega_{1}+I_{1} \omega_{2}}{I_{1}+I_{2}}$ (c) $\frac{I_{1} \omega_{1}-I_{2} \omega_{2}}{I_{1}-I_{2}}$ (d) $\frac{I_{2} \omega_{1}-I_{1} \omega_{2}}{I_{1}-I_{2}}$

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes, rotating with angular frequencies $\omega_{1}$ and $\omega_{2}$ respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be
(a) $\frac{I_{1} \omega_{1}+I_{2} \omega_{2}}{I_{1}+I_{2}}$
(b) $\frac{I_{2} \omega_{1}+I_{1} \omega_{2}}{I_{1}+I_{2}}$
(c) $\frac{I_{1} \omega_{1}-I_{2} \omega_{2}}{I_{1}-I_{2}}$
(d) $\frac{I_{2} \omega_{1}-I_{1} \omega_{2}}{I_{1}-I_{2}}$

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Key Concepts

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in its rotational motion, analogous to mass in linear dynamics. It depends on the mass distribution relative to the axis of rotation, and it plays a crucial role in determining the angular acceleration when a torque is applied.

Inelastic Collision in Rotational Dynamics

When two rotating bodies come into contact and stick together, they undergo an inelastic collision. In such collisions, while angular momentum is conserved, some of the initial rotational kinetic energy is converted into other forms of energy, such as heat, due to friction between the bodies.

Conservation of Angular Momentum

This principle states that in the absence of external torques, the total angular momentum of a system remains constant. When two rotating bodies come into contact and eventually rotate together, their combined angular momentum is equal to the sum of their individual angular momenta before contact.

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