A metal sphere of radius r and specific heat S is rotated...

A metal sphere of radius $r$ and specific heat $S$ is rotated about an axis passing through its centre at a speed of $n$ rotations per second. It is suddenly stopped and $50 \%$ of its energy is used in increasing its temperature. Then, the rise in temperature of the sphere is (a) $\frac{2 \pi^{2} n^{2} r^{2}}{5 S}$ (b) $\frac{\pi^{2} n^{2}}{10 r^{2} S}$ (c) $\frac{7}{8} \pi r^{2} n^{2} S$ (d) $\frac{5(\pi r n)^{2}}{14 S}$

A metal sphere of radius $r$ and specific heat $S$ is rotated about an axis passing through its centre at a speed of $n$ rotations per second. It is suddenly stopped and $50 \%$ of its energy is used in increasing its temperature. Then, the rise in temperature of the sphere is
(a) $\frac{2 \pi^{2} n^{2} r^{2}}{5 S}$
(b) $\frac{\pi^{2} n^{2}}{10 r^{2} S}$
(c) $\frac{7}{8} \pi r^{2} n^{2} S$
(d) $\frac{5(\pi r n)^{2}}{14 S}$

Key Concepts

Rotational Kinetic Energy

This is the energy an object possesses due to its rotation, calculated as (1/2) times the object's moment of inertia (I) multiplied by the square of its angular velocity (?). It plays a key role in problems where mechanical energy is converted into another form, such as heat.

Moment of Inertia

This is a measure of an object's resistance to changes in its rotational motion. For a solid sphere rotating about a diameter, the moment of inertia is given by I = (2/5)mr². It directly influences the calculation of rotational kinetic energy.

Angular Velocity

Angular velocity is the rate at which an object rotates, typically measured in radians per second. It can be derived from the frequency of rotation (in rotations per second) by using the relation ? = 2?n, where n is the number of rotations per second.

Specific Heat

Specific heat is defined as the amount of heat energy required to raise the temperature of one unit mass of a substance by one degree. It is a key parameter in determining how an input of thermal energy translates into a temperature change.

Energy Conversion Efficiency

This concept describes how effectively energy is converted from one form to another. In problems where only a fraction of the available energy is used for heating (e.g., 50% of the kinetic energy being converted into thermal energy), this efficiency directly affects the calculation of temperature increase.

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