Newton's Second Law in Angular Form — Question 3

Question 3

Signed torque area can reverse angular momentum before the pulse ends

Initially $L_z=-1\,\mathrm{kg\,m^2/s}$. About a fixed origin, $\tau_z=2t\,\mathrm{N\,m}$ for $0\le t<2\,\mathrm s$ and $\tau_z=-1\,\mathrm{N\,m}$ for $2\le t\le4\,\mathrm s$.

1. Find final $L_z$ and every zero of $L_z$ during the pulse.

2. Explain why negative torque after 2 s does not immediately imply negative angular momentum.

Question 3 - Solution

Signed torque area can reverse angular momentum before the pulse ends

(a) Derivation and calculation

Angular impulse determines the accumulated change. For $0\le t\le2$, $$L_z(t)=-1+\int_0^t2u\,du=-1+t^2.$$ Therefore $\boxed{t=1.00\ \mathrm s}$ is a zero, and $L_z(2)=3$. For $2\le t\le4$, $$L_z(t)=3+\int_2^t(-1)\,du=5-t,$$ $$\boxed{L_z(4)=1.00\ \mathrm{kg\,m^2/s}}.$$

(b) Result and interpretation

The second-interval formula would vanish at $t=5$ s, outside the stated pulse. Thus the only zero is at 1 s. After 2 s, $$\tau_z=-1<0,\qquad L_z=5-t>0\quad(2\le t\le4).$$ Negative torque gives a negative slope of angular momentum, not necessarily a negative angular momentum.

Physical check

The positive angular impulse is 4 and the negative impulse is $-2$, giving net change $+2$ from the initial $-1$.