OscillationsPeriodic and Oscillatory Motions
Some motions happen only once: a ball rolling to a stop, a stone dropped into a well. Others repeat themselves at regular intervals of time, and these are called periodic motions. The graphs here show three of them (Fig. 13.1): the height above the ground of Sina climbing a ladder and whizzing down a slide, of someone doing a step-up exercise, and of a ball bounced between a palm and the floor.
Choose a motion, then drag the purple knob along its graph to the first moment the motion starts over.
The purple bracket now spans one repeat of the blue graph. In the ball's graph the curved pieces are parabolas, because between the palm and the floor the ball moves with constant acceleration, . Different motions, but each one comes back to exactly the same state after a fixed time.
To and Fro About a Mean Position
Many periodic motions have an equilibrium position somewhere inside their path: a place where no net force acts on the body. Left there at rest, the body would stay there for ever. Give it a small push, though, and a force appears that tries to bring it back. A marble in a bowl is the classic case.
Pull the ball up the side of the bowl and let it go.
The orange arrow is the part of the ball's weight that acts along the bowl. On either side it points back to the bottom, the mean position, and it grows the further the ball is pulled out. So the ball overshoots, is pulled back, overshoots the other way, and so on: a to and fro motion about a mean position is an oscillatory motion.
Real oscillations die away, as yours did, because friction and air resistance take energy out of them. This is called damping. A push from outside at the right moments can keep an oscillation going, which is exactly what the jhulawala does. Whether we call it an oscillation or a vibration is mostly a matter of speed: a branch swaying slowly in the wind oscillates, while a guitar string buzzing hundreds of times a second vibrates.
Periodic, Oscillatory or Neither?
Every oscillation repeats itself, so every oscillatory motion is periodic. The reverse is not true: the Earth goes round the Sun again and again, but it never goes to and fro about a mean position. Sort each motion.
Note
The simplest oscillation of all, and the one this chapter is built on, is simple harmonic motion. It happens when the force on the body is proportional to its displacement from the mean position and always points back towards it. When many oscillators sit side by side, each nudging its neighbours, a disturbance passes along from one to the next: that is a wave, the subject of the next chapter. Sound, ripples on a pond and the shaking of an earthquake all travel this way.
Period and Frequency
The period T is the smallest time after which a motion repeats itself, and its SI unit is the second. Some periods are tiny: the quartz crystal in a digital watch vibrates with a period of a few microseconds (1 µs = 10⁻⁶ s). Others are huge: the planet Mercury takes 88 Earth days to go round the Sun, and Halley's comet comes back every 76 years.
The number of repetitions in each second is the frequency, written ν (the Greek letter nu). One repetition takes T seconds, so
Its unit, s⁻¹, is called the hertz (Hz) after the German physicist Heinrich Rudolf Hertz (1857–1894), who first made radio waves. One hertz is one oscillation per second:
A frequency need not be a whole number.
Drag ν and complete the missions.
Frequency ν =
The purple bracket always spans one period, and the amber dots mark the start of each oscillation. Whatever you set, T × ν = 1.
Example 13.1
Kabir runs up four flights of stairs to his classroom, and his smartwatch counts 96 heartbeats in one minute. What are the frequency and the period of his heartbeat?
His heart beats with a frequency of
Solution
At rest a heart beats about 72 times a minute, a period of about 0.83 s.
Displacement
Earlier, displacement meant the change in a particle's position. In this chapter the word is used more widely: it is the change, with time, of whatever quantity is oscillating, usually measured from its equilibrium value. For a block on a spring the natural choice is the block's distance from its equilibrium point, its displacement x. For a swinging pendulum it is easier to use the angle θ of the string from the vertical (Fig. 13.2).
Fig. 13.2 — (a) a block on a spring on a smooth floor, displaced by x from its equilibrium position; (b) a simple pendulum, displaced by an angle θ from the vertical.
A displacement variable can be almost anything that swings either side of a middle value: the voltage across a capacitor in an AC circuit, the change in air pressure as a sound passes, or the electric and magnetic fields in a light wave. It can be positive or negative, and in an experiment we record it at a series of times.
What plays the part of the displacement for the sound of a tabla reaching your ear?
As the sound passes, the air pressure at your ear rises a little above normal, falls a little below, and back again, hundreds of times a second. That pressure change is the displacement variable of a sound wave.
Sine and Cosine, the Building Blocks
A periodic displacement can be written as a function of time that repeats itself. One of the simplest is
Increase ωt by 2π and the cosine comes back to the same value, so the function repeats after a time T for which ωT = 2π:
and f(t) = f(t + T). A sine function, A sin ωt, repeats in the same way, and so does any mixture of the two:
What does such a mixture look like? Here ω = π rad/s, so every curve repeats every 2 s.
Set A and B and watch the sum.
A =
The teal sine part and the amber cosine part always add up to a purple curve that is itself a single sine wave with the same period, only taller and shifted. To see why, write A = D cos φ and B = D sin φ. Then A sin ωt + B cos ωt = D (sin ωt cos φ + cos ωt sin φ), which is
Fourier's Recipe
Sines and cosines matter so much because of a remarkable result proved by the French mathematician
Switch on the terms one by one.
Each new grey term is smaller and faster than the one before, and with every term the purple sum creeps closer to the dashed square wave. This is why simple harmonic motion, built from a single sine or cosine, is the starting point for understanding every oscillation.
Example 13.2
Which of these functions of time represent periodic motion, and what is the period of each periodic one? (ω is any positive constant.) Each answer draws its function, with ω = π rad/s.
(i) is
(ii) is
(iii) is
(iv) is
Solution
(i) By Eq. (13.3d) this is , a single sine wave, so it is periodic with period 2π/ω.
(ii) The three terms repeat after 2π/ω, π/ω and π/2ω. The longest of these, 2π/ω, is a whole number of times each of the others, so after 2π/ω all three are back where they started together. The sum is periodic with period 2π/ω.
(iii) ω²t² grows steadily and never comes back to an earlier value, so it is not periodic.
(iv) The sine keeps swinging, but every swing is smaller than the last because the factor keeps shrinking. The function never repeats exactly, so it is not periodic. This is what a damped oscillation looks like.
Speed Round: Periodic Motion
ν = 1/T. A motion is periodic only if it comes back to exactly the same state after a fixed time.
Get five right to move on.