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NEET PHYSICSRotational motionMedium

Question

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω1\omega_1 and ω2\omega_2. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is:

A

12I(ω1+ω2)2\frac{1}{2}I(\omega_1+\omega_2)^2

B

14I(ω1−ω2)2\frac{1}{4}I(\omega_1-\omega_2)^2

C

I(ω1−ω2)2I(\omega_1-\omega_2)^2

D

18I(ω1−ω2)2\frac{1}{8}I(\omega_1-\omega_2)^2

Step-by-Step Solution

Let the moment of inertia of each disc be II. According to the law of conservation of angular momentum, the total initial angular momentum equals the total final angular momentum (since no external torque acts on the system): Li=LfL_i = L_f Iω1+Iω2=(I+I)ωcommonI\omega_1 + I\omega_2 = (I + I)\omega_{common} ωcommon=ω1+ω22\omega_{common} = \frac{\omega_1 + \omega_2}{2}

The initial kinetic energy of the system is: Ki=12Iω12+12Iω22K_i = \frac{1}{2}I\omega_1^2 + \frac{1}{2}I\omega_2^2

The final kinetic energy of the system is: Kf=12(2I)ωcommon2=I(ω1+ω22)2=14I(ω1+ω2)2K_f = \frac{1}{2}(2I)\omega_{common}^2 = I \left(\frac{\omega_1 + \omega_2}{2}\right)^2 = \frac{1}{4}I(\omega_1 + \omega_2)^2

The loss of kinetic energy (ΔK\Delta K) is: ΔK=Ki−Kf=(12Iω12+12Iω22)−14I(ω1+ω2)2\Delta K = K_i - K_f = \left(\frac{1}{2}I\omega_1^2 + \frac{1}{2}I\omega_2^2\right) - \frac{1}{4}I(\omega_1 + \omega_2)^2 ΔK=14I[2ω12+2ω22−(ω12+ω22+2ω1ω2)]\Delta K = \frac{1}{4}I [2\omega_1^2 + 2\omega_2^2 - (\omega_1^2 + \omega_2^2 + 2\omega_1\omega_2)] ΔK=14I(ω12+ω22−2ω1ω2)\Delta K = \frac{1}{4}I (\omega_1^2 + \omega_2^2 - 2\omega_1\omega_2) ΔK=14I(ω1−ω2)2\Delta K = \frac{1}{4}I(\omega_1 - \omega_2)^2

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from Rotational motion. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSRotational motionmomentinertiarotatingregularpassing

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