Induction and Energy Transfers — Question 6

Question 6

Increasing circuit resistance with travel changes the braking distance

A coasting rod of mass m starts at x=0 with speed v0. Its circuit resistance grows as $R(x)=R_0+\alpha x$, with alpha positive. Let $C=B^2\ell^2$. Ignore inductance and friction.

1. Find speed as a function of position.

2. Find the limiting distance and recover the constant-resistance limit as alpha tends to zero.

Question 6 - Solution

Increasing circuit resistance with travel changes the braking distance

(a) Method and calculation

$$mv\frac{dv}{dx}=-\frac{Cv}{R_0+\alpha x}\Rightarrow m\frac{dv}{dx}=-\frac C{R_0+\alpha x}.$$$$\boxed{v(x)=v_0-\frac{C}{m\alpha}\ln(1+\alpha x/R_0)}.$$

(b) Interpretation and further calculation

$$v(x_\infty)=0\Rightarrow\boxed{x_\infty=\frac{R_0}{\alpha}\left(e^{m\alpha v_0/C}-1\right)}.$$$$\alpha\to0:\quad e^{m\alpha v_0/C}-1\simeq m\alpha v_0/C\Rightarrow x_\infty\to mR_0v_0/C.$$

Physical check

The cancellation of v in the differential equation is used while v is positive; the final zero-speed position is approached asymptotically.