Two discs of moment of inertia I1 and I2 and angular speeds ω1 and ω2 are rotating along colline

step 1 Dear students, solution for problem 36 of chapter 7 is solution by law of conservation of angula

Two discs of moment of inertia I1 and I2 and angular speeds...

Two discs of moment of inertia $I_{1}$ and $I_{2}$ and angular speeds $\omega_{1}$ and $\omega_{2}$ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate together along the same axis the rotational $K E$ of system will be (a) $\frac{I_{1} \omega_{1}+I_{2} \omega_{2}}{2\left(I_{1}+I_{2}\right)}$ (b) $\frac{\left(I_{1}+I_{2}\right)\left(\omega_{1}+\omega_{2}\right)^{2}}{2}$ (c) $\frac{\left(I_{1} \omega_{1}+I_{2} \omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$ (d) None of these

Two discs of moment of inertia $I_{1}$ and $I_{2}$ and angular speeds $\omega_{1}$ and $\omega_{2}$ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate together along the same axis the rotational $K E$ of system will be
(a) $\frac{I_{1} \omega_{1}+I_{2} \omega_{2}}{2\left(I_{1}+I_{2}\right)}$
(b) $\frac{\left(I_{1}+I_{2}\right)\left(\omega_{1}+\omega_{2}\right)^{2}}{2}$
(c) $\frac{\left(I_{1} \omega_{1}+I_{2} \omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$
(d) None of these

Key Concepts

Conservation of Angular Momentum

This principle 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 contexts where rotating bodies combine or collide, conservation of angular momentum is used to determine the resulting common angular speed of the merged system.

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by a rotating object and is given by the formula (1/2)I?², where I is the moment of inertia and ? is the angular speed. Understanding this energy is crucial when comparing the kinetic energies before and after rotational interactions, even when energy is not necessarily conserved in inelastic processes.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specified axis. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. This concept is essential when analyzing systems where multiple bodies are rotating and are later combined into a single entity.

Combination of Rotating Bodies

When multiple rotating objects are combined to rotate as a single unit, their individual moments of inertia and angular momenta contribute to the final state of the system. This process often involves an inelastic interaction where conservation of angular momentum is applied to determine the resultant angular speed and subsequent kinetic energy of the combined system.

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