The oblong copper plate swings between the poles of an electromagnet. Start the pendulum oscillating, and then turn on the power supply. Eddy currents generated by the motion of the plate through the magnetic field now damp the motion of the pendulum. You can increase or decrease the damping by adjusting the voltage of the power supply and thereby changing the current flowing through the magnet. By adjusting the current appropriately, you can show overdamping, critical damping and underdamping.
The apparatus in the photograph is a pendulum, whose bob is a long rectangular copper plate. The plate rides between the poles pieces of an electromagnet, which is fed by the power supply sitting on the middle shelf of the cart. With the power off, there is no magnetic field, and when you raise the copper plate and then release it, it swings freely between the pole pieces. When you turn on the power supply, there is a magnetic field between the pole pieces of the magnet. With the magnet connected as it is in the apparatus, the north pole happens to be on the left, and the south pole is on right, which means that the field points from left to right. If you raise the copper plate behind the apparatus, as it swings toward the front, the magnetic field induces an EMF that causes (positive) charges to flow upward. (Relative to the magnetic field, the charges in the copper plate are moving toward the front of the cart, and with B pointing to the right, they experience a force, F = qv × B, which points up.) The regions in front of and behind the magnetic field provide return paths for the upward-flowing charges, and this produces two sets of currents, one in front of, and one behind the pole pieces. If we view the apparatus from the left, the currents in front of the magnet flow clockwise, and those behind the magnet flow counterclockwise. Such currents induced in an extended object (as opposed to a ring or a loop of wire) are called eddy currents.
As the pendulum bob swings forward, the charges flowing upward in the magnetic field experience a force, F = qv × B, which points toward the rear of the apparatus, in the opposite direction to the motion of the pendulum bob. The other parts of the eddy currents, which are outside the field, do not experience such a force, and there is a net backward force on the pendulum bob, which greatly slows it or stops it. Similarly, if you raise the pendulum from the front (or if it happens to pass through the bottom of its swing and be returning on its backswing), in the region between the magnet pole pieces the magnetic field induces an EMF that causes (positive) charges to flow downward. (v is now reversed, so qv × B points downward.) This produces two sets of eddy currents that flow in opposite directions to those produced when the pendulum swings forward. The charges flowing downward in the magnetic field now experience a force, F = qv × B, which points toward the front of the apparatus, again in the opposite direction to the motion of the pendulum bob. This, again, greatly slows or stops the pendulum bob. This phenomenon is also known as eddy current damping or magnetic damping. It is perhaps most commonly used in mechanical balances, in which an aluminum vane attached to the end of the beam rides within a slot in the vertical member that holds the zero indicator. Magnets on either side of the slot induce eddy currents in the vane as the beam oscillates up and down, which slows the oscillation and reduces its amplitude so that the beam comes to rest more quickly than it otherwise would.
If the magnetic field in the pendulum apparatus pointed in the opposite direction, the motions of the currents in the middle of the field would be in opposite directions to those described above (as would the associated sets of eddy currents), but since the field now points from left to right, qv × B would still oppose the motion of the pendulum bob. We see that this damping force is proportional to the (tangential) velocity of the pendulum.
Because the copper plate swings on an arm that has significant mass (and the plate is an extended object), this pendulum is a physical pendulum, as opposed to a simple pendulum, which in the ideal case comprises a point mass at the end of a massless string. (The behavior of the simple pendulum is described on the page for demonstration 40.21 -- Pendulums of different lengths and masses.) Therefore, we must analyze its motion in terms of the restoring torque that gravity provides when we displace the pendulum from equilibrium. If we call the distance between the pivot and the center of mass of the pendulum d, and the mass of the pendulum M, then if we displace the pendulum bob to one side at an angle of θ, the torque that gravity exerts to restore the pendulum to its equilibrium position is
τ = -Mgd sin θ
For small angles, sin θ ≈ θ, and for small pendulum swings τ ≈ -Mgd θ. If we call mgd κ, then we have τ = -κ θ. This torque also equals the angular acceleration of the pendulum, or τ = Iα = I (d2θ/dt2), so that I (d2θ/dt2) = -κ θ, (d2θ/dt2) = -(κ/I) θ, and
(d2θ/dt2) + (κ/I) θ = 0.
This is similar to the equations for the oscillating mass-spring and the simple pendulum, except that it is in θ instead of x, and the constant on the second term is different. Its solution gives
θ = θ0 cos(ωt - φ)
where θ0 is the initial (maximum) displacement of the pendulum bob, ω is the angular frequency of the pendulum and φ is a phase factor that depends on the boundary conditions. ω = √(κ/I). The period, T, equals 2π/ω, and T = 2π√(I/κ) = 2π√(I/Mgd). If we consider the special case of a point mass, m, suspended from the end of a (massless) string of length l, then I = ml2, M = m, and d = l, and T = 2π√(I/Mgd) = 2π√(l/g).
When you turn the magnet on, this provides a damping force that is linearly related to the speed of the pendulum bob, which is d (dθ/dt), where d is the distance between the pivot and the center of the magnet pole faces. The resulting equation is:
(d2θ/dt2) + cd(dθ/dt) + (κ/I) θ = 0.
The characteristic equation that corresponds to this is r2 + cdr + κ/I = 0. The roots to this equation are [-cd ±√(c2d2 - 4κ/I)]/2. If we substitute 2b for cd, and substitute ω2 for κ/I, the equation becomes
(dθ2/dt2) + 2b(dθ/dt) + ω2θ = 0,
and the roots become r1 = -b + √(b2 - ω2) and r2 = -b - √(b2 - ω2). This gives rise to three different situations, depending on the relative sizes of b and ω:
1) If b2 - ω2 > 0, the system is said to be overdamped (or overcritically damped). The roots are real and unequal, and both are negative. The solution of the differential equation above is:
θ = C1er1t + C2er2t.
Since both r1 and r2 are negative, θ approaches zero as time increases. Depending on the conditions, the mass may cross the equilibrium position once, but no more than once, and the system does not oscillate. Demonstration 40.33 -- Mass-springs damped in oil, water, shows an overdamped system (the mass-spring damped in oil).
2) If b2 - ω2 = 0, the system is said to be critically damped. The roots are equal, and the solution of the differential equation is:
θ = (C1t + C2)e-ωt (or θ = (C1t + C2)e-bt).
As time increases, θ approaches zero (but does not cross it). (The mass also approaches the equilibium position more quickly than it does when the system is overdamped.) Demonstration 40.33 -- Mass-springs damped in oil, water, shows a critically-damped system (the mass-spring damped in water).
3) If b2 - ω2 < 0, the system is said to be underdamped. In this case, the roots are complex and unequal. If we let ω2 - b2 = α2, the roots are r1 = -b + αi and r2 = -b - αi. This leads to the solution
θ = e-bt(C1 cos αt + C2 sin αt).
If we set C1 = A cos φ and C2 = A sin φ (where A = √(C12 + C22) and tan φ = C2/C1), we can use the trigonometric identity
A cos(αt - φ) = A cos φ cos αt + A sin φ sin αt
to obtain
θ = Ae-bt cos (αt - φ)
This equation describes damped oscillation. This is similar to simple harmonic motion whose frequency, in radians per second, is α, and whose period is T = 2π/α, except that the amplitude is not constant. Because of the factor e-bt, which is called the damping factor, the amplitude (which equals Ae-bt), decays over time. As noted above, b, in the damping factor, equals cd/2. The greater the damping force, the larger b is, and the more quickly the oscillations decay. We can see from the equation above, that the displacement of the pendulum bob oscillates between the two curves x = Ae-bt and x = -Ae-bt.
The period of oscillation, T = 2π/α = 2π/√(ω2 - b2), is longer than that for the system in which there is no damping, for which T = 2π/ω. We can see that as, and b becomes smaller, T approaches that for the undamped system. Also, the damping factor approaches unity, and the equation above reduces to θ = A cos (ωt - φ). If we take A = θ0, this is the equation for the undamped pendulum, given above.
With no current flowing through the magnet, the pendulum in this demonstration is underdamped. When you turn the magnet on, you add damping. If you start with low current, the pendulum is still underdamped. As you increase the current, the damping increases, and at some point, the pendulum is critically damped. If you increase the current further, the pendulum is overdamped.
References:
1) Sears, Francis Weston and Zemansky, Mark W. College Physics, Third Edition (Reading, Massachusetts: Addison-Wesley Publishing Company, 1960) p. 672-3.
2) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition(New York: John Wiley and Sons, 1977), pp. 313-314, 788.
3) Thomas, George B., Jr. and Finney, Ross L. Calculus and Analytic Geometry (Reading, Massachusetts: Addison-Wesley Publishing Company, 1992), pp. 1081-1082.