Angular Momentum — Question 5

Question 5

A spiral trajectory need not conserve angular momentum

A particle of mass $m$ follows planar polar coordinates $r=r_0(1+t/t_0)$ and $\theta=t/t_0$, where $r_0,t_0>0$.

1. Find its angular momentum about the polar origin.

2. Compare $L$ at $t=t_0$ and at $t=0$, and identify which velocity component contributes.

Question 5 - Solution

A spiral trajectory need not conserve angular momentum

(a) Derivation and calculation

Polar velocity is $$\vec v=\dot r\hat e_r+r\dot\theta\hat e_\theta.$$ Since $\hat e_r\times\hat e_\theta=\hat z$, $$\vec L=mr\hat e_r\times\vec v=mr^2\dot\theta\hat z.$$ With $\dot\theta=1/t_0$, $$\boxed{L_z(t)=\frac{mr_0^2}{t_0}(1+t/t_0)^2}.$$

(b) Result and interpretation

At the two stated times, $$L_z(0)=mr_0^2/t_0,\qquad L_z(t_0)=4mr_0^2/t_0,$$ $$\boxed{L_z(t_0)/L_z(0)=4}.$$ The radial velocity has zero direct cross product with $\vec r$, but the changing radius affects both the tangential speed $r\dot\theta$ and its lever arm.

Physical check

Doubling radius at fixed angular speed multiplies angular momentum by four.