A video of this demonstration is available at this link.
A 500-g mass (blue) hangs on a (vertical) string that is attached to the same (horizontal) string from which seven other pendulums are suspended. Each of the other pendulums is made of a paper (oak tag) cone, and all of them have different lengths. You can hang the blue mass on its string at any of three positions. When you set it oscillating, this sets the other pendulums in motion. All except one have different length, and thus different natural oscillation frequency, from that of the blue pendulum. These move at the same frequency as the blue pendulum, but with a lag whose size depends on the relationship between their natural frequency and that of the blue pendulum, and their motion is small (or becomes small over time). The pendulum whose length, and therefore natural frequency, matches that of the pendulum with the blue mass, begins to oscillate with great motion, with a 90° phase lag with respect to the motion of the blue pendulum. One can observe this phase lag with the apparatus as shown above, but it is more obvious if you turn the table 90 degrees to this orientation.
The blue mass at left in the photograph above, hangs from a string that is attached to a horizontal string from which seven other pendulums, all of different length, and whose bobs are made of stiff paper cones. This arrangement is also known as Barton’s pendulum. The pendulum with the blue mass is thus coupled to all the other pendulums. When you set it oscillating, through the common string it exerts a periodic force on all the other pendulums. The page for demonstration 40.21 -- Pendulums of different lengths and masses, gives the differential equation that describes the motion of a pendulum, with its solution. If we include damping, and we assume that this damping is proportional to the speed of the pendulum, this equation is:
m(d2x)/(dt2) + c(dx/dt) + (mg/l)x = 0.
If we add the driving force, we have:
m(d2x)/(dt2) + c(dx/dt) + (mg/l)x = F0 cos ωt,
where F0 cos ωt = FD(t), the periodic driving force. If we divide through by m, define the damping factor γ = c/2m, and note that the natural frequency of the pendulum is ω0 = √(g/l), this yields the solution
x = Ae-γt sin (ω′t + φ) + ((F0/m)/[(ω02 - ω2) + 4γ2ω2]1/2) cos (ωt - δ).
The first term on the right, which describes the motion of a damped, free oscillator, decays over time, and is thus called the transient part of the solution. The amplitude factor A equals (x0 - C cos δ)/sin φ, and ω′ = √(ω02 - γ2). The second term on the right does not decay with time, and is called the steady-state part of the solution. We can see that if the damping is large, the amplitude of oscillation is small. Conversely, if the damping is small, the amplitude can be large, depending on the other factors in the equation. We also see that if the frequency of the driving force is much different from the natural oscillating frequency of the pendulum, the amplitude of the motion is small, and that as the driving frequency approaches the natural oscillating frequency, the amplitude increases. The phase factor δ gives the relationship – the lag – between the driving force and the motion of the pendulum, and it equals tan-1 [2γω/(ω02 - ω2)].
If we plot the amplitude of the steady-state term as a function of the driving frequency, the curve we obtain is low at points far from ω0 on either side, and rises from either side as ω approaches ω0, to reach a maximum at ω0. The breadth of the peak – its full width at half maximum – at ω0 depends on the damping, γ. For small γ, this width equals γ. A quantity denoted by Q, which equals ω0/γ, is a measure of this. (This quantity is often called the quality.) The higher Q is, the narrower the resonance peak, and the nearer ω must be to ω0 for the amplitude to be large. Thus, after the system has been oscillating for some time, only the pendulum whose natural frequency of oscillation matches that of the pendulum with the blue mass, exhibits motion of significant amplitude, while the motion of all the other pendulums is relatively small.
We also see from the phase factor δ, that if the driving frequency is much lower than the natural frequency of oscillation of the pendulum, the motion of the pendulum is almost in phase with that of the driving pendulum (δ ≈ 0). As ω increases, the phase angle, and thus the lag between the motion of the driving pendulum and that of the driven pendulum, slowly increases. (Since δ is subtracted from ωt, it is a negative phase angle.) As ω approaches ω0, the phase angle increases more and more quickly. When ω = ω0, δ = 90°; the motion of the driven pendulum lags that of the driving pendulum by one quarter cycle. As ω increases further, δ continues to increase. When ω is no longer close to ω0, δ inreases more and more slowly. When ω greatly exceeds ω0, δ approaches 180°.
When you set the pendulum with the blue mass in motion, the pendulum whose length, and thus oscillating frequency, matches that of the pendulum with the blue mass, begins to swing in a wide arc. The motion of the other pendulums either dies down after being large at the start, or is always small. It is fairly easy to see the 90° lag between the blue pendulum and the excited pendulum, especially with the apparatus turned sideways. Though the motion of the other pendulums is fairly small, it is also possible to observe the phase difference between their motion and that of the blue mass. For those pendulums whose natural frequency is lower than that of the blue mass, the lag is greater than 90°, and for those whose natural frequency is higher than that of the blue mass, the lag is smaller than 90°.
References:
1) Braun, Martin. Differential Equations and Their Applications (New York: Springer-Verlag, 1978) pp. 161-164.
2) Feynman, Richard P., Leighton, Robert B. and Sands, Matthew. The Feynman Lectures on Physics, Volume I (Reading, Massachusetts: Addison-Wesley Publishing Company, 1963) pp. 23-3 to 23-5.
3) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 323-325.
4) http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html#c1 and sections and links below.
5) http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr2.html#c1 and links therein/sections below.
6) Prof. Frank Krauss, in the physics department at Durham University (UK), has a page on damped and driven pendulums here.