Induction and Energy Transfers — Question 7

Question 7

Two moving rods dissipate relative kinetic energy while preserving their common motion

Two conducting rods of equal mass m slide freely on the same rails and close a rectangular circuit. Their velocities are v1 and v2 along x. Rail spacing is ell and the field is perpendicular. Total resistance R is constant; neglect inductance and rail forces along x.

1. Find the decay law of relative velocity u=v2-v1.

2. If initially v1=0 and v2=v0, find final common speed and total heat.

Question 7 - Solution

Two moving rods dissipate relative kinetic energy while preserving their common motion

(a) Method and calculation

$$i=\frac{B\ell(v_2-v_1)}R,\quad m\dot v_2=-ku,\quad m\dot v_1=+ku,\quad k=B^2\ell^2/R.$$$$\dot u=-2ku/m\Rightarrow\boxed{u=u_0e^{-2kt/m}}.$$

(b) Interpretation and further calculation

$$v_1+v_2=v_0\Rightarrow\boxed{v_f=v_0/2}.$$$$H=\tfrac12mv_0^2-2[\tfrac12m(v_0/2)^2]=\boxed{mv_0^2/4}.$$

Physical check

Induction damps separation changes. Rigid common translation preserves flux and survives after the relative motion has decayed.