Question

Calculate the rotational inertia of a wheel that has a kinetic energy of 24400 Jwhen rotating at 602rev/min.

Short answer

Rotational inertia of a wheel is, l = 12.3 kg.m².

Step-by-step solution

Understanding the given information

1. Kinetic energy of the wheel is, K = 24400 J.
2. Angular velocity of the wheel is, $\omega =602rev/min$.

Concept and Formula used in the given question

By using the concept of Rotational kinetic energy, we can find rotational inertia of thewheel.

Rotational Kinetic energy$\mathit{K}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{l}{\mathit{\omega }}^{\mathbf{2}}$

Calculation for the rotational inertia of a wheel

To calculate rotational inertia of the wheel we will use rotational kinetic energy formula and rearrange it for inertia I

$\begin{array}{rcl}K& =& \frac{1}{2}l{\omega }^2\\ l& =& \frac{2K}{{\omega }^2}\end{array}$ …(1)

We have K = 24000 J

$\omega =602rev/min$

Converting it into rad/sec

$\begin{array}{rcl}\omega & =& 602rev/min\times \frac{2\mathrm{\pi rad}/\mathrm{rev}}{60s/min}\\ & =& 63rad/s\end{array}$

Using $\omega$and K in equation (1), we get

$\begin{array}{rcl}l& =& \frac{2\times 24400J}{{\left(63rad/s\right)}^2}\\ & =& 12.29kg.m^2\\ & \approx & 12.3kg.m^2\end{array}$

Hence the rotational inertia of a wheel (I) is 12.3kg.m².