Work and Rotational Kinetic Energy — Question 8

Question 8

A torque-speed characteristic has a maximum-power point

A motor's delivered torque is $\tau(\omega)=10-0.50\omega\,\mathrm{N\,m}$ for $0\le\omega\le20\,\mathrm{rad/s}$.

1. Find the angular speed of maximum output power and that power.

2. Compare power at stall and at the zero-torque speed. Why is the largest torque not the largest power?

Question 8 - Solution

A torque-speed characteristic has a maximum-power point

(a) Derivation and calculation

Multiply torque by angular speed before optimizing: $$P(\omega)=(10-0.50\omega)\omega=10\omega-0.50\omega^2.$$ $$\frac{dP}{d\omega}=10-\omega=0\Rightarrow\boxed{\omega_*=10\ \mathrm{rad/s}}.$$ $$\frac{d^2P}{d\omega^2}=-1<0,\qquad \boxed{P_{\max}=10(10)-0.50(10)^2=50\ \mathrm W}.$$

(b) Result and interpretation

At the two endpoints, $$P(0)=10(0)=\boxed{0},\qquad P(20)=(10-10)(20)=\boxed{0}.$$ Stall has the largest torque but no motion; the zero-torque rate has motion but no driving moment. Maximum power needs both factors to be nonzero.

Physical check

The maximum occurs halfway between the two zeros of the concave quadratic.