Periodic and oscillatory motion
Motions that repeat
Look around you. The hands of a clock go round and round. The Earth spins once a day, and the Moon circles the Earth about once a month. Your heart beats again and again.
Each of these motions repeats itself after a fixed interval of time. A motion that repeats itself at regular intervals of time is called periodic motion.
Now watch a child on a swing in the park, a boat bobbing on a river, and the string of a sitar after it is plucked. These move back and forth about a middle position.
That middle position is the equilibrium, or mean, position. If the body is left there at rest, it stays there, because the net external force on it is zero.
Pull the swing a little to one side and let go. A force pulls it back towards the mean position. It overshoots, comes back, and the to-and-fro motion repeats.
A periodic to-and-fro motion about a mean position is called oscillatory motion. When the oscillations are very fast, like a tuning fork or a guitar string, we usually call them vibrations.
Here is the key idea. Every oscillatory motion is periodic, but not every periodic motion is oscillatory. A ceiling fan blade comes back to the same place every turn, but it never moves to and fro. Its motion is periodic, not oscillatory.
Some motions are not periodic at all. A cart sliding to a stop on a rough road never repeats anything. In this chapter, we study the simplest and most important oscillation: simple harmonic motion.
Period, frequency and displacement
The smallest interval of time after which a motion repeats itself is called its period, T. Its SI unit is the second.
Periods cover a huge range. A quartz crystal in a watch vibrates in microseconds. Mercury takes 88 days to go round the Sun, and Halley’s comet comes back about every 76 years.
The number of repetitions per unit time is the frequency, written ν or f. It is simply one over the period: ν equals 1 over T.
Frequency is measured in hertz, after Heinrich Hertz. One hertz is one oscillation per second. A frequency need not be a whole number.
Worked example. On average, a human heart beats 75 times in one minute. The frequency is 75 divided by 60 seconds, which is 1.25 hertz. The period is 1 over 1.25, or 0.8 seconds.
To describe an oscillation, we use the displacement: how far the quantity has changed from its mean value. For a block on a spring, it is the distance from the equilibrium position, and it can be positive or negative.
Displacement need not be a distance. For a pendulum, it is the angle from the vertical. In other oscillations it can be a voltage, a pressure, or an electric field.
We can write the displacement as a function of time. For example, f of t equals A cos ωt. Cosine repeats every 2π radians, so this repeats when ωT equals 2π. The period is T equals 2π over ω.
Add sines and cosines with periods T, T over 2 and T over 4, and the result still repeats every T. But a function like e to the power minus ωt only falls and never repeats. It is not periodic.
Try it: which motions repeat?
- Choose each motion in turn and press Play (or Space).
- The shaded band marks one period. Does the graph after it look exactly the same?
- The fan is periodic but never goes to and fro. The cart sliding to a stop never repeats.
- Change T and watch f = 1/T change in the readouts.
Periodic, but not oscillatory
Which of these motions is periodic but NOT oscillatory?
- The pendulum of a wall clock
- A child on a swing
- A blade of a ceiling fan turning at a steady speed
- A block on a spring on a smooth floor
Answer
C. The fan blade returns to the same position every turn, so its motion is periodic. But it never moves back and forth about a mean position, so it is not oscillatory. The other three swing to and fro about a mean position: they are both periodic and oscillatory.
Heartbeat
A student’s heart beats 72 times in one minute. What is the period of the heartbeat?
- 0.72 s
- 0.83 s
- 1.2 s
- 72 s
Answer
B. Frequency ν = 72 beats ÷ 60 s = 1.2 Hz. The period is T = 1/ν = 1/1.2 = 0.83 s. The answer 1.2 s mixes up the frequency (in hertz) with the period (in seconds).
Summary
- Frequency:
ν = 1/T - Unit of frequency:
1 Hz = 1 s⁻¹ - Periodic function:
f(t + T) = f(t) - Period of A cos ωt:
T = 2π/ω
- A motion that repeats at regular intervals of time is periodic. The smallest such interval is the period T.
- A periodic to-and-fro motion about a mean (equilibrium) position is oscillatory. Every oscillation is periodic, but circular motion is periodic without being oscillatory.
- Frequency is the number of oscillations per second, ν = 1/T, measured in hertz.
- Displacement is the change from the mean value: a distance, an angle, a voltage or a pressure. It can be positive or negative.
- A sum of sines and cosines with periods T, T/2, T/3… repeats every T. A decaying function like e−ωt is not periodic.