A uniform disc of mass m and radius r can rotate freely about its axis, which is horizontal and

step 1 So in this question, we have a uniform disk of mass small m and radius small r which can freely

A uniform disc of mass m and radius r can rotate freely...

A uniform disc of mass $\mathrm{m}$ and radius $\mathrm{r}$ can rotate freely about its axis, which is horizontal and passing through its centre. A point mass $\mathrm{m}$ is attached at the edge of the disc at the same level as the center. The maximum angular velocity is (a) $\sqrt{\frac{\mathrm{g}}{\mathrm{r}}}$ (b) $\sqrt{\frac{\mathrm{g}}{2 \mathrm{r}}}$ (c) $2 \sqrt{\frac{\mathrm{g}}{3 \mathrm{r}}}$ (d) $\sqrt{\frac{g}{5 r}}$

A uniform disc of mass $\mathrm{m}$ and radius $\mathrm{r}$ can rotate freely about its axis, which is horizontal and passing through its centre. A point mass $\mathrm{m}$ is attached at the edge of the disc at the same level as the center. The maximum angular velocity
is
(a) $\sqrt{\frac{\mathrm{g}}{\mathrm{r}}}$
(b) $\sqrt{\frac{\mathrm{g}}{2 \mathrm{r}}}$
(c) $2 \sqrt{\frac{\mathrm{g}}{3 \mathrm{r}}}$
(d) $\sqrt{\frac{g}{5 r}}$

Key Concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational acceleration about an axis. For composite systems, the total moment of inertia is the sum of the individual moments, calculated with respect to the same axis. In problems involving a disc and an attached point mass, the moment of inertia of each component is computed separately and then summed to capture the overall rotational inertia of the system.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation and is given by (1/2)I?², where I is the moment of inertia and ? is the angular velocity. This concept is crucial in rotational dynamics as it links the rotational motion of a system to its energy, thereby allowing for analysis through energy conservation methods.

Gravitational Potential Energy Change

Gravitational potential energy (GPE) represents the energy stored in an object due to its position in a gravitational field. When an object in a rotating system changes its vertical position, the difference in GPE can be converted into rotational kinetic energy. In dynamics problems where a mass moves under gravity, the change in height determines the amount of energy available for conversion.

Conservation of Energy in Rotational Systems

The conservation of energy principle states that energy cannot be created or destroyed but only transformed from one form to another. In rotational dynamics, this means that the loss in gravitational potential energy by a mass can be equated to the gain in rotational kinetic energy of the system. This relationship is used to derive values like the maximum angular velocity by setting the energy change equal to the rotational kinetic energy.

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