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A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig.). A massless string is wrapped over its rim and two blocks of masses m1 and m2 (m1 > m2) are attached to the ends of the string. The system is released from rest. If the angular speed of the wheel when m1 descends by a distance h is -

1. [2(m1 − m2)gh/((m1 + m2)R² + I)]^(1/2)
2. [2(m1 + m2)gh/((m1 + m2)R² + I)]^(1/2)
3. [(m1 + m2)/((m1 + m2)R² + I)]^(1/2) gh
4. [(m1 − m2)/((m1 + m2)R² + I)]^(1/2) gh

Correct answer: [2(m1 − m2)gh/((m1 + m2)R² + I)]^(1/2)

Solution

The correct option accounts for the net force acting on the system, which is the difference in weights of the two masses (m1 - m2), and incorporates the rotational inertia of the wheel. This results in a derived angular speed that reflects both the translational motion of the masses and the rotational motion of the wheel, ensuring energy conservation is maintained.

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