A solid sphere of radius R, rotating with an angular velocity abo… - Vidyadip

Question

A solid sphere of radius $R$, rotating with an angular velocity $\omega$ about its diameter, suddenly stops rotating and $75 \%$ of its $\mathrm{KE}$ is converted into heat. If $\mathrm{c}$ is the specific heat capacity of the material in SI units, show that the temperature of $3 \mathrm{R}^2 \mathrm{CO}^2$ the sphere rises by $\frac{3 R^2 \omega^2}{20 c}$.

✓

Answer

The $\mathrm{Ml}$ of a solid sphere about its diameters, $\mathrm{I}=\frac{2}{5} \mathrm{MR}^2$ where $M$ is its mass.
The rotational KE of the sphere,
$
\begin{aligned}
E=\frac{1}{2} I \omega^2 & =\frac{1}{2}\left(\frac{2}{5} M R^2\right) \omega^2 \\
& =\frac{1}{5} M R^2 \omega^2
\end{aligned}
$
If $\Delta \theta$ is the rise in temperature,
$
\begin{aligned}
& \quad M c \Delta \theta=\frac{3}{4} E=\frac{3}{4}\left(\frac{1}{5} M R^2 \omega^2\right) \\
& \therefore \Delta \theta=\frac{3 R^2 \omega^2}{20 c}
\end{aligned}
$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Answer the following in Brief" from Rotational Dynamics→Rotational Dynamics — all question types→Physics STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Derive an expression for the net torque on a rectangular current carrying loop placed in a uniform magnetic field with its rotational axis perpendicular to the field.

Assuming expression for impedance in a parallel resonant circuit, state the conditions for parallel resonance. Define resonant frequency and obtain an expression for it.

A spherical drop of oil falls at a constant speed of $9.8 cm / s$ in steady air. Calculate the radius of the drop. The density of oil is $0.9013 g / cm ^3$, density of air is $0.0013 g / cm ^3$ and the coefficient of viscosity of air is $1.8 \times 10^{-4}$ poise.

Draw a neat labelled schematic diagram of transferring heat from a cold region to a hot region.

A coil has $1000$ turns, each of area $0.5 m ^2$. What is the magnetic moment of the coil when it carries a current of $1 \ mA$ ?

What are the steps through which a refrigerant goes in one complete cycle of refrigeration?

An alternating emf is applied to a series combination of an inductor and a resistor $(R=100\Omega).$ If the impedance of the circuit is $100 \sqrt{2} \Omega,$ what is the phase difference between the emf and the current?

A stone of mass $100 \mathrm{~g}$ attached to a string of length $50 \mathrm{~cm}$ is whirled in a vertical circle by giving it a velocity of $7 \mathrm{~m} / \mathrm{s}$ at the lowest point. Find the velocity at the highest point.

Derive an expression for orbital magnetic moment of an electron revolving around the nucleus in an atom. State the formula for the Bohr magneton.

A torque of $20 \mathrm{~N}$.m sets a stationary circular disc into rotation about a transverse axis through its centre and acts for $2 \pi$ seconds. If the disc has a mass $10 \mathrm{~kg}$ and radius $0.2 \mathrm{~m}$, what is its frequency of rotation after $2 \pi$ seconds ?