Magnet falling through a copper tube: forces, energy and a slit
A cylindrical neodymium magnet, 10 mm in diameter and of mass 12.0 g, is dropped from rest into a vertical copper tube 1.00 m long with a 14 mm bore. Phone slow-motion video gives fall times of 2.35 s, 2.40 s and 2.45 s. The same magnet dropped through a cardboard tube of the same length takes 0.45 s. The department then supplies an identical copper tube with a 2 mm wide slit cut along its entire length; the magnet now takes 0.52 s. In discussion, one student says the magnet falls slowly in copper because copper is not magnetic, so it is being repelled; a second says it should fall faster in copper because magnetic materials are attracted to metals; a third says the magnet has reached terminal velocity, so the acceleration is zero and the forces must balance, which proves that gravity is being cancelled. The teacher asks the group to test these statements against the data and to account for the energy.
1. Estimate the average magnetic force on the magnet during the 2.40 s fall and compare it with the magnet's weight. State the assumptions in your estimate.
Read the worked answer
Average acceleration from rest: a = 2s/t² = 2.00 / 5.76 = 0.347 m s⁻².
Weight: W = mg = 0.0120 × 9.81 = 0.118 N. Net force: ma = 0.0120 × 0.347 = 4.2 × 10⁻³ N, so the average magnetic force is W − ma = 0.118 − 0.004 = 0.114 N, about 96 % of the weight.
Assumptions and checks: (i) constant average acceleration - the real drag grows with speed, so 0.114 N is a time-average, not the force at any instant; (ii) air resistance is negligible - the cardboard-tube time of 0.45 s corresponds to a = 2.00/0.2025 = 9.88 m s⁻², a shortfall of only 0.7 % from g; (iii) the magnet does not touch the tube walls; (iv) the average speed is 1.00/2.40 = 0.42 m s⁻², roughly half of what free fall would give over the same time (4.4 m s⁻¹), confirming a strong, speed-dependent braking force.
2. Explain why the magnet does not simply accelerate at a constant reduced rate, and say where the gravitational potential energy goes.
Read the worked answer
The induced emf is proportional to the rate of change of flux through each ring of copper, which is proportional to the magnet's speed v. The induced (eddy) current is emf/R, also proportional to v, and the force on the magnet from that current is proportional to Iv, hence to v. So the equation of motion is a = g − kv/m: the acceleration starts near g and falls as v grows, and the magnet approaches a terminal speed v_t = mg/k, of the order of the observed 0.42–0.5 m s⁻¹. The motion is therefore not constant acceleration at all - the 0.347 m s⁻² figure is only a time average.
Energy: mgh = 0.0120 × 9.81 × 1.00 = 0.118 J. At a typical 0.45 m s⁻¹ the kinetic energy is ½ × 0.0120 × 0.2025 = 1.2 × 10⁻³ J, about 1 % of the drop. Essentially all the lost potential energy appears as Joule heating in the copper (plus a little in the magnet), not as kinetic energy.
3. Account for the 0.52 s fall time in the slitted tube, and identify which of the three students' statements are wrong and why.
Read the worked answer
The eddy currents circulate in closed loops around the tube axis, in planes roughly perpendicular to the magnet's motion. A full-length longitudinal slit breaks every one of those loops, turning each ring into a nearly open C-shape and raising the resistance by orders of magnitude. The induced current, and hence the braking force, collapses, so the magnet falls almost freely (0.52 s against 0.45 s). The remaining 0.07 s excess is the residual current that can still circulate in the narrow C-section. This is a controlled comparison that isolates the induced-current mechanism from air resistance, mass or geometry.
Student 1: the mechanism is not repulsion from a non-magnetic metal; the force comes from induced currents and by Lenz's law always opposes the motion. Student 2: copper is not ferromagnetic, so there is no attraction, and attraction towards the tube walls would not speed the fall anyway. Student 3: reaching terminal velocity is correct, but 'gravity is cancelled' is wrong - gravity is unchanged and is balanced by the magnetic drag, and the net force is zero only at terminal speed.