Question

A ring of radius 0.5 m and mass 10 kg is rotating about its diameter with angular velocity of 20 rad/s. Its KE is :-
A. 10J
B. 100J
C. 500J
D. 250J

Answer 1

A rigid body's moment of inertia is a number that defines the torque required to achieve a desired angular acceleration along a rotating axis, similar to how mass influences the force required to achieve a desired acceleration. It relies on the mass distribution of the body and the axis selected, with greater moments necessitating more torque to affect the rate of rotation. We use this concept here to solve the problem.

Formula used:
${E_K} = \dfrac{1}{2}I{\omega ^2}$
I = moment of inertia
$\omega$= angular velocity

Complete step by step solution:
The kinetic energy generated by an object's rotation, also known as angular kinetic energy, is a component of its overall kinetic energy. When you look at rotational energy independently around an object's axis of rotation, you'll see the following dependency on the object's moment of inertia.
${E_{{\text{rotational}}}} = \dfrac{1}{2}I{\omega ^2}$
The torque times the rotation angle equals the mechanical work needed or exerted during rotation. The torque times the angular velocity is the instantaneous power of an angularly accelerated body. The axis of rotation for free-floating (unattached) objects is usually centred on their centre of mass.
Moment of inertia of a ring about its diameter ${\text{I}} = \dfrac{1}{2}{\text{m}}{{\text{r}}^2}$
and kinetic energy is given by ${{\mathbf{E}}_{\text{K}}} = \dfrac{1}{2}{\text{I}}{\omega ^2}$
So from the given inputs of the problems it can be deduced as
${{\mathbf{E}}_{\text{k}}} = \dfrac{1}{4}{\text{m}}{{\text{r}}^2}{\omega ^2}$
$= \dfrac{1}{4} \times 10 \times {(0.5)^2} \times {(20)^2}$
$= {\mathbf{250J}}$
Hence ${E_K} = 250J$.
Therefore the correct option is $\left( D \right)$.

Note:
Note that when I is induced to ${E_K}$, $\dfrac{1}{2}$is getting multiplied with another $\dfrac{1}{2}$to get $\dfrac{1}{4}$.
The object's rotation generates rotational energy, which is a component of its overall kinetic energy. The moment of inertia is seen when the rotational energy is examined independently across an object's axis of rotation. The kinetic energy owing to an object's rotation, also known as angular kinetic energy, is defined as: The kinetic energy due to an object's rotation, which is a portion of its overall kinetic energy.