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

Two discs of moment of inertia $I_{1}$ and $I_{2}$ about their respective axes and rotating with angular speed $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. Then the loss of in kinetic energy of the system in the process is (a) $\frac{l_{1} b}{2\left(I_{1}+b_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$ (b) $-\frac{l_{1} b}{2\left(I_{1}+b_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$ (c) $\frac{l_{1} l_{2}}{\left(I_{1}+l_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$ (d) zero

Two discs of moment of inertia $I_{1}$ and $I_{2}$ about their respective axes and rotating with angular speed $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. Then the loss of in kinetic energy of the system in the process is
(a) $\frac{l_{1} b}{2\left(I_{1}+b_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$
(b) $-\frac{l_{1} b}{2\left(I_{1}+b_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$
(c) $\frac{l_{1} l_{2}}{\left(I_{1}+l_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$
(d) zero

Key Concepts

Angular Momentum Conservation

Angular momentum conservation states that when no external torques act on a system, the total angular momentum before an interaction is equal to the total angular momentum after the interaction. In problems involving rotating bodies coming into contact, such as two discs colliding with friction, this principle is used to determine the final common angular speed after the collision, despite the loss of kinetic energy.

Rotational Kinetic Energy

Rotational kinetic energy is the energy of a rotating body, expressed as one?half the product of its moment of inertia and the square of its angular speed. This concept is crucial in analyzing energy changes during interactions of rotating objects since, unlike linear collisions where kinetic energy might be partially conserved, rotational interactions with friction typically result in energy loss due to inelastic effects.

Moment of Inertia

The moment of inertia is a measure of how the mass of an object is distributed relative to a chosen axis of rotation. It plays a similar role to mass in linear motion but for rotational dynamics. Its value affects both the angular momentum and the rotational kinetic energy of an object, and it is essential in calculations involving the combined rotation of multiple bodies.

Inelastic Collision in Rotational Motion

In an inelastic collision involving rotating bodies, friction between the objects causes them to eventually rotate together at a common angular speed, leading to a loss of kinetic energy. Although the total angular momentum is conserved, the energy dissipated as heat or sound results in a final kinetic energy that is less than the initial total kinetic energy. This energy difference is a fundamental concept when analyzing energy losses during rotational collisions.

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