Newton's Second Law in Angular Form — Question 10

Question 10

Trajectory curvature reconstructs torque without a force diagram

A $2\,\mathrm{kg}$ particle follows $x=t$, $y=t^2$, $z=0$ in numerical SI units relative to a fixed origin.

1. Find $L_z(t)$ and infer torque by differentiation.

2. Independently calculate force from acceleration and verify the torque.

Question 10 - Solution

Trajectory curvature reconstructs torque without a force diagram

(a) Derivation and calculation

Differentiate the trajectory to find $\vec v=(1,2t,0)$. Then $$L_z=m(xv_y-yv_x)=2[t(2t)-t^2(1)] =\boxed{2t^2\ \mathrm{kg\,m^2/s}},$$ $$\boxed{\tau_z=dL_z/dt=4t\ \mathrm{N\,m}}.$$

(b) Result and interpretation

Independently, the acceleration and force are $$\vec a=(0,2,0),\qquad\vec F=m\vec a=(0,4,0)\ \mathrm N.$$ Their moment about the same fixed origin is $$\boxed{(\vec r\times\vec F)_z=t(4)-t^2(0)=4t\ \mathrm{N\,m}}.$$ The force calculation agrees with the angular-momentum derivative.

Physical check

Both methods use the same position origin and inertial momentum, as the torque law requires.