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The Regenerative Rail Brake

EM induction, Faraday/Lenz, LR

Engineering students build a model of regenerative braking for a Wellington-style cable car. A copper rod of mass

m = 0.250\ \text{kg}

rests across two parallel, smooth metal rails a distance

L = 0.400\ \text{m}

apart, with the rod perpendicular to the rails. The rails are

2.00\ \text{m}

long and lie in a plane inclined at

\theta = 10.0^\circ

to the horizontal. Instead of a resistor, the rails are joined at the top by an initially uncharged supercapacitor of capacitance

C = 2.50\ \text{F}

, rated for a maximum p.d. of

2.70\ \text{V}

. The resistance and self-inductance of the rod, rails and capacitor are negligible unless stated otherwise. A uniform magnetic field

B = 0.800\ \text{T}

acts vertically (not perpendicular to the incline). The rod is released from rest at the top of the rails and slides down without friction. The diagram shows the arrangement from the side and looking down onto the plane of the rails.

Data: g = 9.81 m s⁻²

(a)

show that2 marks

Explain why, at every instant, the p.d. across the capacitor equals the emf

BLv\cos\theta

induced in the rod, where

v

is the rod's speed. Hence, by considering the forces acting on the rod, show that it moves down the rails with a constant acceleration

a = \dfrac{mg\sin\theta}{m + CB^2L^2\cos^2\theta}

(b)

calculate2 marks

Calculate the acceleration of the rod and compare it with the acceleration with no magnetic field. Then calculate the speed of the rod and the p.d. across the capacitor when the rod reaches the bottom of the rails, and check whether the capacitor's rating is exceeded.

(c)

show that2 marks

Show that, at any instant, the ratio of the energy stored in the capacitor to the kinetic energy of the rod is

\dfrac{CB^2L^2\cos^2\theta}{m}

. Calculate both energies when the rod reaches the bottom and show that together they account for the gravitational potential energy lost.

(d)

discuss2 marks

In the students' first design the rails were joined by a resistor, and the rod reached a terminal speed. Discuss why the capacitor design gives a constant acceleration instead of a terminal speed, including the limiting cases

C \to 0

and

C \to \infty

. Then discuss how the motion would change if the supercapacitor had a small internal resistance.

Your score: 0 / 8(0 of 4 parts marked — open “Mark it” on each part)