Angular Impulse, Work Done by Torque and Rotational Power

Angular impulse

The rotational twin of $F\Delta t=\Delta p$ is

and, just as with force, when the torque varies the angular impulse is the area under a $\tau$–$t$ graph. A constant $3$ N·m torque applied for $4$ s changes the angular momentum of anything by $12$ kg·m²/s, whatever its shape. Dividing by $I$ then gives $\Delta\omega$ — so the same angular impulse spins a small $I$ up far more than a large one.

A particle moving in a straight line has angular momentum

This is the idea that trips most students. About a chosen point, a particle of mass $m$ and speed $v$ has

where $r_\perp$ is the perpendicular distance from the point to the line of motion. A ball flying past a pivot at $4$ m/s and $1$ m off-line carries $L=m(4)(1)$ about that pivot even though it is not rotating. Because gravity and the pivot force produce no torque about the pivot during a brief collision, this $L$ is what is conserved when the ball hits a pivoted rod and sticks — while linear momentum is not conserved, since the pivot pushes on the rod with an external force.

Work and power of a torque

These are $W=Fd$ and $P=Fv$ with the translation table applied, and $\Delta\theta$ must be in radians. Work done by a torque changes rotational kinetic energy: $\tau\Delta\theta=\Delta\left(\tfrac12I\omega^2\right)$. A motor delivering $20$ N·m at $300$ rpm ($\omega=\frac{2\pi\cdot300}{60}\approx31{,}4$ rad/s) has output power $P=20\times31{,}4\approx628$ W. The same torque at twice the speed is twice the power — which is why engine power ratings quote both a torque and an rpm.