Question:
Why is the energy transfer most efficient when the driver leads the pendulum by a quarter cycle?
Answer:
Firstly, notice that the pendulums end up having different phase relationships with the driver, depending on whether the driving frequency is too high or too low (compared to the resonant frequency)
1) For the two shortest pendulums, because the driving frequency is too low, they end up oscillating roughly in-phase with the driver.
2) For the two longest pendulums, because the driving frequency is too high, they end up oscillating roughly in anti-phase with the driver.
3) For the middle pendulum, because the driving frequency matches its resonant frequency, it ends up oscillating with a quarter of a cycle lag (or PI/2 radians) behind the driver.
So it looks like when the driver leads the pendulum by a quarter cycle, the energy transfer is most efficient. But why is that so?
To understand this, we must study the direction of the periodic driving force. Whenever the driver is displaced to the right, it is pulling the pendulums rightward. Whenever the driver is displaced to the left, it is pulling the pendulums leftward. This means that the driver is exerting a periodic driving force which is in-phase with its own displacement.
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Now, remember that the velocity of a shm leads its displacement by a quarter cycle? What this means is that (see graphs above) if the driver is leading the pendulum by a quarter cycle, the period driving force it is exerting is actually in-phase with the pendulum’s velocity! In other words, the driver is always pulling the pendulum in the direction it is moving. This means that the driver is always doing positive work and inputting energy to the pendulum all the time. This explains why this pendulum goes into resonance. Because the transfer of energy from the driver to the pendulum is at its most efficient.
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On the other hand, if the driver’s is in-phase with the pendulum’s displacement (which is with the short pendulums), it actually means that the periodic driving force is one quarter cycle behind pendulum’s velocity. Look at the graphs above. Do you realize that driver is pulling the pendulum in the same direction as the pendulum is moving only half the time? The other half of the time, the driver is actually pulling the pendulum in opposite direction to the pendulum’s motion, slowing the pendulum down instead. The net work done by the driver on the pendulum (over every complete cycle) is close to zero. No wonder the pendulum oscillates at small amplitudes.
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Now that we can appreciate why the phase relationship determines the efficiency of energy transfer, some students are going to ask why the driver ends up a quarter-cycle ahead of the pendulum during resonance in the first place? Well, that can only be shown using very complicated mathematics, which is beyond the A-level syllabus. In fact, as far as A-level is concerned, all you need to know is that energy transfer is most efficient during resonance.
xmPhysics 