Barton's Pendulum

The actual mathematics of driven oscillations is quite complicated, but doable in 1st year uni, if you want to see exactly how much things are out of phase with the driver.

However, you can understand it without the maths. The 1st reference in that Wikipedia article is probably the best one to look at to understand to begin with. Note that the way it's set up, the driver is much heavier than any of the other pendula, so it's motion is basically not affected, so it dominates the motion of the system, swings at its own frequency, and drives all of the other pendula pretty much independently of one another.

First consider the resonant case (the case of the driver driving the lighter pendulum of the same length). This pendulum likes swinging at the same frequency as the heavier one. The best example I was told was when you are pushing someone on a swing, you are driving their oscillation. To push most effectively, you apply your force just as they are reaching the top of their swing.

If you think of your motion as an oscillation too (your body moves backwards and forwards to push the swing), when their displacement is a maxmium (top of swing), your displacement is zero (your velocity is maximum - you're moving through the middle of your "oscillation" in order to push them with the most speed.) The difference between maximum displacement and zero displacement in an oscillation is pi/2 (or T/4), so the swing is T/4 out of phase with you (the driver).

Now imagine you're holding a weight on the end of a string. If you move your hand at the resonant frequency, then the weight will resonate like a swing and its amplitude will increase.

If you try to move your hand very much slower than the resonant frequency, all that will happen is that the weight follows your movement, thus it will be in phase.

If you try to move your hand quicker than the resonant frequency, then when the weight is maximum displaced to the left, your hand will move very quickly to the right and the weight can't respond that fast, so it will move over to the right too, but by the time it's there, your hand will have moved back to the left again. Thus, you are T/2 out of phase. The greater your frequency compared to the resonant frequency, the lower the amplitude of the driven oscillator. Eventually if you drive too fast, the driven component will have close to zero amplitude.

So the pendulum that is the same length as the driver has the same frequency, and thus resonates T/4 out of phase (like the driver is 'pushing' the pendulum's "swing")

The shorter pendula have much higher resonant frequencies than the driver, and so just follow in phase with the driver. Their amplitudes will be different depending on how far away they are from the driver frequency.

The longer pendula have much lower