Induction and Energy Transfers — Question 4

Question 4

A constant applied force initially builds kinetic energy before heating dominates

A rod on resistive rails starts from rest under constant horizontal force F0. Set $k=B^2\ell^2/R$ and neglect inductance and friction.

1. Find speed and the fraction of instantaneous input power converted to heat.

2. Find the time when heating and kinetic-energy growth receive equal power.

Question 4 - Solution

A constant applied force initially builds kinetic energy before heating dominates

(a) Method and calculation

$$m\dot v=F_0-kv\Rightarrow v=\frac{F_0}{k}(1-e^{-t/\tau}),\quad\tau=m/k.$$$$\boxed{\frac{P_R}{P_{\mathrm{in}}}=\frac{kv^2}{F_0v}=1-e^{-t/\tau}}\quad(t>0).$$

(b) Interpretation and further calculation

$$P_R=\dot K\Rightarrow P_R/P_{\mathrm{in}}=1/2.$$$$\boxed{t=\tau\ln2}.$$

Physical check

At t=0 both powers vanish, so the ratio there is interpreted as a right-hand limit.