Newton's Second Law in Angular Form — Question 2

Question 2

A measured vector history identifies torque and its effect on magnitude

Angular momentum about a fixed origin is $\vec L(t)=(2t,3t^2,4)$ in numerical SI units.

1. Find torque at $t=1\,\mathrm s$.

2. Find the instantaneous rate of change of $|\vec L|$ at that time.

Question 2 - Solution

A measured vector history identifies torque and its effect on magnitude

(a) Derivation and calculation

Differentiate each component: $$\vec L(t)=(2t,3t^2,4)\Rightarrow\vec\tau(t)=(2,6t,0).$$ Thus $$\boxed{\vec\tau(1)=(2,6,0)\ \mathrm{N\,m}},\qquad\vec L(1)=(2,3,4).$$

(b) Result and interpretation

Differentiate $L^2=\vec L\cdot\vec L$: $$2L\dot L=2\vec L\cdot\vec\tau\Rightarrow\dot L=\frac{\vec L\cdot\vec\tau}{L}.$$ At $t=1$ s, $$\boxed{\dot L=\frac{2(2)+3(6)+4(0)}{\sqrt{2^2+3^2+4^2}} =\frac{22}{\sqrt{29}}=4.085\ \mathrm{N\,m}}.$$ The perpendicular torque component changes direction rather than magnitude instantaneously.

Physical check

The magnitude-change rate is smaller than $|\vec\tau|=\sqrt{40}$ N m because the torque is not parallel to $\vec L$.