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A solid disc with radius a is connected to a spring at a point d above the center of the disc. The other end of the spring is fixed to the vertical wall. The disc is free to roll without slipping on the ground. The mass of the disc is M and the spring constant is K. The polar moment of inertia for the disc about its centre is J = Ma²/2. The natural frequency of this system in rad/s is given by

1. √(2K(a+d)² / 3Ma²)
2. √(2K / 3M)
3. √(2K(a+d)² / Ma²)
4. √(K(a+d)² / Ma²)

Correct answer: √(2K(a+d)² / 3Ma²)

Solution

The correct option is derived from the dynamics of the rolling disc and the spring-mass system, where the effective moment of inertia and the spring's restoring force contribute to the natural frequency. The factor of 2 in the numerator accounts for both translational and rotational motion, while the division by 3M reflects the distribution of mass and the geometry of the system.

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