Oscillation and the Pendulum - Physics for Pilot Assessments

Every wave in this module was made by something oscillating, so it is worth looking at oscillation on its own. By the end of this lesson you should be able to say what makes a system oscillate rather than simply move, explain why a pendulum keeps almost perfect time whatever you hang on it, and work out what happens to its period when you change its length — including the fact that four times the length gives only twice the period.

Oscillation about an equilibrium position

An oscillation is a repeated back-and-forth motion about one particular position, and that position has a name: the equilibrium position, the place the object sits when it is left alone and nothing is disturbing it. For a pendulum it is hanging straight down; for a mass on a spring it is the point where the spring's pull exactly balances the weight; for a ruler clamped to a desk it is the undeflected position. Everything an oscillation does is described relative to that position, and the vocabulary is the same as for waves, because a wave is what you get when oscillations are passed from one place to the next.

So the amplitude of an oscillation is the greatest distance it reaches from the equilibrium position — not from one extreme to the other, exactly as with a wave. Its period, written T, is the time for one complete cycle, and one complete cycle means all the way there and all the way back, so a pendulum's period is the time from one extreme right round to the same extreme again, not the time from one side to the other. Its frequency is the number of complete cycles per second, in hertz, and the two are reciprocals of one another exactly as before: the period is 1 divided by the frequency, and the frequency is 1 divided by the period.

The restoring force

What makes something oscillate rather than simply move away and stay there is the direction of the force acting on it. Displace it from equilibrium and there must be a force pushing or pulling it back towards equilibrium — a restoring force — and typically the force grows the further it is displaced. A stretched spring is the clearest case: from Hooke's law in the matter and fluids module, the force is proportional to the extension and it always pulls back towards the natural length. Pull a pendulum bob to one side and the same thing happens in a different geometry: gravity pulls straight down, the string stops it going down, and what is left over is a net force pulling the bob back towards the lowest point, growing as the bob is displaced further.

A restoring force alone would only bring the object back to equilibrium and stop it there. What makes the motion continue past that point is energy, from the conservation lesson earlier in this module. As the bob falls towards the lowest point it converts stored energy into movement, so it arrives at equilibrium at its fastest — and an object moving at speed does not stop simply because the force on it has momentarily fallen to zero. It overshoots, and now it is on the other side, where the restoring force acts the other way, slowing it down until it stops, at which point the whole exchange runs in reverse. A restoring force plus a store of energy is what an oscillation is made of.

One cycle of an oscillation, seen as an energy exchange. Height is traded for speed on the way down and speed for height on the way up, and the restoring force is what turns the bob round at each end.

Why the mass of the bob makes no difference

Hang a heavy bob and a light bob on strings of the same length, set them going together, and they keep step indefinitely. This is genuinely surprising the first time, and the reason is one you have already met in the mechanics module. A heavier bob does have a larger restoring force pulling it back, in proportion to its mass — but it also has more mass to be accelerated by that force, in exactly the same proportion. From F = ma, the acceleration is the force divided by the mass, so if the force doubles when the mass doubles then the acceleration is unchanged. Both bobs follow the same path at the same speeds, and they take the same time about it. It is the same cancellation that makes all objects fall at the same rate when air resistance is negligible.

The amplitude makes no appreciable difference either, provided the swings are small. A pendulum pulled a long way to the side has further to travel, but it also builds up more speed on the way down, and for modest angles the two effects cancel almost exactly — so a swing that is dying away keeps almost exactly the same period all the way through, which is what made pendulums the basis of accurate clocks for three centuries. That statement carries an honest limit: for large swings, beyond roughly twenty degrees from the vertical, the cancellation stops being clean and the period does creep up a little with amplitude. The tests treat pendulums as small-swing, and so does this lesson.

What the period does depend on

Two things: the length of the pendulum, and the strength of gravity. A longer pendulum has a longer period — it swings more slowly — and the reason is geometrical rather than mysterious: the bob on a long string traces a much longer arc for the same sideways displacement, while the restoring force pulling it along that arc is no greater, so the journey takes longer. Weaker gravity has the same effect: the same pendulum on the Moon, where the gravitational field strength is about a sixth of the Earth's, would swing markedly more slowly, because the restoring force at any displacement is smaller.

The important part is how the period depends on the length, because it is not proportional. The period grows with the square root of the length, the square root of a number being the value which multiplied by itself gives that number.

How a pendulum period scales with length

T \propto \sqrt{L}

$T$: the period — the time for one complete swing there and back, in seconds (s)
$\propto$: is proportional to — multiply the right-hand side by any factor and the left is multiplied by the same factor
$L$: the length of the pendulum, in metres (m)
$\sqrt{\ }$: the square root — the number which, multiplied by itself, gives what is inside

Read that as a recipe for scaling rather than as something to calculate with. Four times the length means the square root of four times the period, and the square root of 4 is 2, so the period doubles. Nine times the length gives three times the period. Going the other way, a quarter of the length gives half the period, because the square root of a quarter is a half. The full expression behind that proportionality, for completeness, brings in gravity and a factor of two pi.

The period of a simple pendulum, in full

T = 2 \pi \sqrt{\frac{L}{g}}

$T$: the period, in seconds (s)
$\pi$: pi, about 3.14 — a bare number with no units
$L$: the length of the pendulum, in metres (m)
$g$: gravitational field strength, about 10 newtons per kilogram (N/kg) on Earth

You will not be asked to evaluate that under time pressure, and it is quoted here so that nothing in the lesson has to be taken on trust. What it confirms is everything said above: the length is under a square root, gravity is under the same square root and on the bottom so more gravity means a shorter period, and the mass of the bob appears nowhere at all. The one number worth carrying from it is that a pendulum one metre long has a period of very nearly two seconds on Earth, which is why the pendulum in a longcase clock is about that length and ticks once a second, once for each half of its swing.

Why it matters in aviation

Aircraft have natural oscillations of their own. Disturb one in pitch and it does not simply settle: it exchanges height for speed and back again in a slow cycle lasting tens of seconds, rising a little as it slows and sinking as it regains speed. How quickly that cycle dies away is a real part of how an aircraft handles, and a pilot who tries to correct every swing by hand can end up reinforcing it, since a badly timed input adds energy to the oscillation exactly as a well-timed push adds energy to a swing.

The pendulum has a more concrete legacy in the instruments. Anything mounted so that it is free to swing responds to acceleration as well as to gravity, because it hangs along the combination of the two rather than along the vertical. A magnetic compass card is suspended in just that way, which is why its indication cannot be trusted while the aircraft is accelerating, decelerating or turning, and why a heading is read from it only once the aircraft has settled into steady flight.

What the test asks

The dominant distractor is the heavy bob: an item offers two pendulums of the same length with different masses and invites you to say the heavier one swings more slowly, on the reasonable-sounding grounds that it is harder to move. The mass cancels, and any option that makes the period depend on the bob is wrong. The second trap is proportionality: given four times the length, the option four times the period is offered alongside the correct two times, and it is chosen by everybody who did not take the square root. Expect too an item testing what one complete cycle means, where the trap counts a single one-way swing as a period rather than the full journey there and back, and one that changes the amplitude and expects the period to change with it, when for small swings it does not.