Kinetic Energy of Rotation — Question 6

Question 6

Energy fractions and speed fractions are not interchangeable

A fixed-inertia flywheel initially has angular speed $80\,\mathrm{rad/s}$. A device extracts $75\%$ of its kinetic energy without reversing it.

1. Find the final speed.

2. If the initial speed is instead reduced by $75\%$, what fraction of energy is extracted?

Question 6 - Solution

Energy fractions and speed fractions are not interchangeable

(a) Derivation and calculation

Removing 75 percent of the energy leaves one quarter: $$\frac{K_f}{K_i}=1-0.75=0.25=\left(\frac{\omega_f}{\omega_i}\right)^2.$$ With no reversal, $$\boxed{\omega_f=80\sqrt{0.25}=40\ \mathrm{rad/s}}.$$

(b) Result and interpretation

A 75 percent speed reduction instead gives $$\frac{\omega_f}{\omega_i}=0.25,\quad\frac{K_f}{K_i}=(0.25)^2=\frac1{16},\qquad \boxed{\frac{K_i-K_f}{K_i}=\frac{15}{16}=93.75\%}.$$

Physical check

Halving speed removes three quarters of the energy; reducing speed to one quarter removes still more.