Question

If rotational kinetic energy is $50\%$ of total linear kinetic energy then the body will be

Answer 1

Step 1: Given data

Rotational kinetic energy is $50\%$ of total linear kinetic energy.

1. The energy of moving objects is known as kinetic energy. This motion might be either linear or rotating. As a result, an object in rotation will also have kinetic energy.
2. Rotational kinetic energy is the name given to this type of energy. The kinetic energy of a rotating object is determined by its angular velocity, which is measured in radians per second.
3. It also depends on the moment of inertia of the item. The moment of inertia is a measurement of how easy it is to change an object's rotating state

Step2: Formulas used

$K{E}_{\text{linear}}=\frac{1}{2}m{v}^2$

where $m$ is the mass and $v$ is the velocity

$K{E}_{\text{rotational}}=\frac{1}{2}I{\omega }^2$

where $I$ is the oment fo inertia along the axis of rotation and $\omega$ is the angular velocity

Step 3: Calculation

We have,

$\begin{array}{ccc}& K{E}_{\text{rotational}}=0.5K{E}_{\text{linear}}& \\ \Rightarrow & \frac{1}{2}I{\omega }^2=\left(0.5\right)\frac{1}{2}m{v}^2& \\ \Rightarrow & I{\omega }^2=\frac{1}{2}m{\left(\omega r\right)}^2& \left[\because v=\omega r\right]\\ \Rightarrow & I=\frac{m{r}^2}{2}& \end{array}$

The moment of inertia about the axis of rotation for uniform circular disc is $I=\frac{m{r}^2}{2}$

Demonstrating with the help of figure.

Hence, the body will be a uniform circular disc if its rotational kinetic energy accounts for $50\%$ of total linear kinetic energy.