A man is standing at the centre of a big flat disc which is...

A man is standing at the centre of a big flat disc which is rotating with angular speed $\omega .$ The man starts running with ac* apration $a$ with respect to disc. If $M, m, I, R$ are mass of disc, mass of man, M.I. of disc and radius of disc respectively, the angular acceleration of disc when man is at a distance $x$ from centre is : (a) $\frac{I \omega}{\left(1+m x^{2}\right)} 2 m x \sqrt{2} a x$ (b) $\frac{I \omega}{\left(1+m x^{2}\right)^{2}} \sqrt{2} a x$ (c) $\frac{I \omega}{\left(1+m x^{2}\right)^{2}} 2 m x \sqrt{2} a x$ (d) none of these

A man is standing at the centre of a big flat disc which is rotating with angular speed $\omega .$ The man starts running with ac* apration $a$ with respect to disc. If $M, m, I, R$ are mass of disc, mass of man, M.I. of disc and radius of disc respectively, the angular acceleration of disc when man is at a distance $x$ from centre is :
(a) $\frac{I \omega}{\left(1+m x^{2}\right)} 2 m x \sqrt{2} a x$
(b) $\frac{I \omega}{\left(1+m x^{2}\right)^{2}} \sqrt{2} a x$
(c) $\frac{I \omega}{\left(1+m x^{2}\right)^{2}} 2 m x \sqrt{2} a x$
(d) none of these

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Key Concepts

Conservation of Angular Momentum

In a closed system with negligible external torques, the total angular momentum must remain constant. When the man accelerates relative to the disc, the internal forces produce equal and opposite angular momentum changes in the man and the disc. This principle governs how the disc responds by undergoing an angular acceleration that counters the man's change in angular momentum.

Moment of Inertia

The moment of inertia measures an object's resistance to changes in its rotational motion, depending on the mass distribution around the axis of rotation. In the problem, both the disc's moment of inertia and the man’s contribution (modeled as a point mass at a distance from the center) are crucial for determining the system's rotational response when the man accelerates.

Relative Motion in Rotating Systems

When an object moves with acceleration relative to a rotating frame, its motion must be analyzed by considering both the acceleration in the rotating frame and the induced reactive effects on the system. The man's acceleration relative to the disc causes changes in the mechanical configuration of the system that lead to a compensating angular acceleration of the disc.

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