# 043 Barton’s Pendulum (Phase Relationship)

Do notice that the pendulums have different phase relationships with the driver.

1) The two shorter pendulums oscillate roughly in-phase with the driver.

2) The two longer pendulums oscillate roughly completely out-of-phase with the driver.

3) The pendulum in resonance lags the driver by about a quarter of a cycle (or PI/2 radians). This kind of makes sense. Because if the driver’s displacement leads the pendulum’s displacement by a quarter of a cycle, it actually means that the driver’s displacement is in-sync with the pendulum’s velocity. This means the driver is always tugging the pendulum in the same direction as the pendulum’s velocity1. This means the driver is in the position to be doing positive work and inputting energy to the pendulum all the time. The pendulum goes into resonance because the transfer of energy from the driver to the pendulum is at its most efficient. On the other hand, if the driver’s displacement is in-sync with the pendulum’s displacement, it actually means that the driver’s displacement is one quarter cycle behind pendulum’s velocity. This means the driver is tugging the pendulum in the same direction as the pendulum’s velocity only half the time. The other half of the time, the driver is actually tugging the pendulum in the opposite direction as the pendullum’s velocity, slowing the pendulum down. The net work done by the driver on the pendulum is close to zero. No wonder the pendulum oscillate at small amplitudes.

The same thing happens if the driver is completely out of phase with the pendulum. It can be shown using complicated mathematics that the phase relationship between the driver and the forced oscillation depends on the mismatch between the driving frequency and the resonant frequency. This goes in some way to explain the shape of the resonance curve.

1. Note that whenever the driver is displaced to the right, it is tugging the pendulums to the right. Likewise, whenever the driver is displaced to the left, it is tugging the pendulums to the left. In other words, the the periodic driving force is in-sync with the displacement of the driver.