Induction and Energy Transfers — Question 2

Question 2

A freely coasting rod stops only asymptotically but travels a finite distance

A rod of mass m on frictionless closed rails starts at speed v0. The circuit resistance is R, rail spacing ell and perpendicular field B. Self-inductance is negligible.

1. Find velocity and the limiting travel distance.

2. Find the total heat released by the time the speed has fallen to v0/2.

Question 2 - Solution

A freely coasting rod stops only asymptotically but travels a finite distance

(a) Method and calculation

$$m\dot v=-\frac{B^2\ell^2}{R}v,\quad\tau=\frac{mR}{B^2\ell^2}\Rightarrow\boxed{v=v_0e^{-t/\tau}}.$$$$x(t)=v_0\tau(1-e^{-t/\tau})\Rightarrow\boxed{x_\infty=v_0\tau}.$$

(b) Interpretation and further calculation

$$H=K_0-K=\frac12m[v_0^2-(v_0/2)^2].$$$$\boxed{H=\frac38mv_0^2=\frac34K_0}.$$

Physical check

Finite total travel does not imply stopping at a finite time in the ideal linear-drag model.