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Stopping the Zipline

EM induction, Faraday/Lenz, LR

A zipline at an adventure park near Rotorua uses an eddy-current brake. A pair of permanent magnets is mounted on the trolley, one pole on each side of a stationary vertical aluminium fin that runs along the horizontal braking section of the track. The fin has thickness

d = 6.0\ \text{mm}

and height

10\ \text{cm}

, and is longer than the braking distance, so the magnets stay over the fin until the rider stops. In a simple model, each pole face is a square of side

a = h = 5.0\ \text{cm}

, centred on the fin's height; the field between the poles is uniform with

B = 0.40\ \text{T}

perpendicular to the fin, and zero elsewhere. The rider and trolley have total mass

m = 90\ \text{kg}

and enter the braking section at

v_0 = 15\ \text{m s}^{-1}

. Ignore friction and air resistance.

Data: Conductivity of aluminium σ = 3.5 × 10⁷ S m⁻¹ · Density of aluminium ρ = 2.7 × 10³ kg m⁻³ · Specific heat capacity of aluminium c = 9.0 × 10² J kg⁻¹ K⁻¹ · Temperature coefficient of resistivity of aluminium α = 4.0 × 10⁻³ K⁻¹

(a)

show that2 marks

Treat the part of the fin between the poles as a conducting block of height

h

, length

a

(along the track) and thickness

d

, moving at speed

v

relative to the magnets. Assume the induced current crosses the block vertically, along

h

, and returns through the fin outside the field with negligible resistance. Show that the size of the braking force is

F = \sigma B^2vahd

, and explain how this force acts on the trolley even though the fin is fixed to the track.

(b)

calculate1 mark

Calculate the braking force and the deceleration (in multiples of

g

) as the rider enters the brake.

(c)

derive2 marks

Using the model

F = kv

, and by considering the change in the rider's momentum during a short time

\Delta t

in which the rider moves

\Delta x = v\Delta t

, derive an expression for the stopping distance and calculate it. Comment on the time taken to stop according to this model.

(d)

estimate1 mark

The heat is produced in the stretch of fin that is between the poles at each moment. Estimate the largest temperature rise produced in the fin by one rider, stating your assumptions.

(e)

explain2 marks

The model in (a) overestimates the braking force. Explain two physical reasons why. After a busy, hot afternoon the fin is

40\ \text{K}

hotter than when the brake was calibrated. Explain, in terms of the conduction electrons in the aluminium, why riders now stop further along the track, and calculate the new stopping distance.

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