An old pendulum clock hangs in the corridor of your school. Its minute hand goes round the dial once an hour, again and again. Below the dial, the pendulum swings left and right, once every second or so. Both motions repeat, and both keep the clock's time. Yet they are clearly different. One goes round, and the other goes to and fro.
Motion that repeats
Many motions repeat themselves at regular intervals of time. The Earth goes round the Sun once a year. The minute hand of a clock goes round once an hour. A stone whirled on a string at a steady speed is back at the same point after every turn. Such a motion is called periodic motion. The fixed interval after which it repeats is its period, \(T\).
In symbols, a motion is periodic if the position at any time \(t\) equals the position one period later. The velocity and the acceleration then repeat as well. A motion that repeats after \(T\) also repeats after \(2T\), \(3T\) and so on. The period is the smallest such time.
A motion is periodic if it repeats itself exactly after a fixed interval of time. The smallest such interval is the period \(T\), and the position obeys \(x(t + T) = x(t)\) at every instant \(t\).
A wheel that goes round but never repeats
Not every motion that looks regular is periodic. Follow one point on the rim of the wheel of a moving cycle. The point goes round the axle once in every turn of the wheel. But the wheel also moves forward along the road. After one turn the point is back at the road, a full circumference further on.
Look at the readout in Fig 2.1. Seen from the road, P is never again at the point where it started. Its distance from the start keeps growing, so its motion never repeats, and the motion is not periodic. Seen from the axle, the same point simply goes round a circle. There the motion repeats every \(4\,\text{s}\). Like rest and motion, periodicity depends on the frame of reference.
A car moves along a straight road at a steady speed. Is the motion of a point on the rim of one of its wheels periodic, seen from the road?
To and fro
Some periodic motions go round, like the minute hand. Others go back and forth along the same path, like the pendulum of the clock. A periodic to-and-fro motion about a fixed point is called an oscillation, or a vibration. The fixed point is the mean position. The body swings past it in both directions and turns back at two extreme positions.
A swing in a park, a mass bobbing on a spring and the prongs of a tuning fork all oscillate. So does the needle of a sewing machine and the air in a flute. In each case the body speeds up as it moves towards the mean position and slows down as it moves away. At each extreme it stops for an instant and turns back.
What is the mean position of an oscillation?
Where oscillations come from
Why do so many bodies oscillate? Recall the chapter on Potential Energy and Conservation of Energy. A body at a position of stable equilibrium feels no net force. Displace it a little, and a force pushes it back towards that position. The body arrives there with some speed, overshoots, and the force now pushes it back from the other side. The result is an oscillation about the position of stable equilibrium.
So the mean position of an oscillation is a position of stable equilibrium. A marble in a steel katori rolls to and fro about the bottom. A marble on an upturned katori does not oscillate. It rolls away, because the top is an unstable equilibrium.
Every oscillation is periodic, because the to-and-fro cycle repeats in a fixed time. The reverse is not true. The Earth's orbit and the minute hand are periodic, but neither goes to and fro, so neither is an oscillation.
A real swing slowly loses its motion, and each swing is a little smaller than the last. Strictly, such a motion does not repeat exactly, so it is not periodic. In this chapter every oscillation is ideal: nothing takes energy from it, and it repeats for ever. Losses are the subject of the chapter Superposition, Damped and Forced Oscillations.
Which statement about ideal oscillations and periodic motions is correct?
Which of these is not a feature of an oscillation?
The simplest oscillation
Oscillations come in many shapes. The simplest one is also the most common. In it, the acceleration of the body is proportional to its displacement from the mean position, and always points back towards that position. Measure the displacement \(x\) along the line of motion, from the mean position. Then the acceleration is
\[ a = -\omega^2 x \]
Here \(\omega\) is a positive constant of the motion, called the angular frequency. Its unit is rad/s, or simply s⁻¹, because \(\omega^2 x\) must be an acceleration. This motion is called simple harmonic motion, or SHM for short. The word simple means that the law is the simplest possible, a straight proportion. The word harmonic comes from music, because the strings of instruments move this way.
A particle moves in simple harmonic motion when its acceleration is proportional to its displacement from a fixed point. The acceleration always points towards that point: \(a = -\omega^2 x\). The fixed point is the mean position, and the constant \(\omega\) is the angular frequency.
What the minus sign does
The minus sign carries the whole idea. When the particle is on the positive side, \(x > 0\), the acceleration is negative, towards the mean position. On the negative side it is positive, again towards the mean position. The farther the particle is from the mean position, the larger the acceleration. At the mean position itself, \(x = 0\) and the acceleration is zero.
Fig 2.2 shows both arrows at once. At the ends, \(x = \pm 4\,\text{cm}\), the particle stops for an instant, yet its acceleration is largest, \(16\,\text{cm/s}^2\). At O the particle is fastest, yet its acceleration is zero. The graph below the line shows the law itself. Every state of the motion lies on one straight line through the origin, whose slope is \(-\omega^2 = -4\,\text{s}^{-2}\).
The two arrows in Fig 2.2 are never largest together. The acceleration is largest at the ends, where it reverses the motion. The velocity is largest at the mean position, which the particle reaches after being pulled towards it for a quarter of the cycle.
This chapter describes SHM, and does not ask what produces it. Something must supply the acceleration, such as a spring, gravity on a pendulum, or the pressure of a gas. The next chapter, Dynamics of SHM, finds the force \(F = -kx\) behind each case. Here \(a = -\omega^2 x\) is simply the definition of the motion.
In SHM the two extremes lie at the same distance from the mean position, one on each side. That distance is called the amplitude, \(A\). In one full cycle the particle goes from the mean position to \(+A\), back to the mean position, on to \(-A\) and back again. It covers the distance \(A\) four times, so it travels \(4A\) in every period. Its displacement over a full period is zero.
A particle moves along the \(x\) axis. Positions are in metres and accelerations in m/s². Which of these accelerations give SHM, and with what \(\omega\) and mean position? (a) \(a = -9x\) (b) \(a = +9x\) (c) \(a = -9x^3\) (d) \(a = 18 - 9x\)
(a) SHM. The acceleration is \(-\omega^2 x\) with \(\omega^2 = 9\), so \(\omega = 3\,\text{rad/s}\), about the mean position \(x = 0\).
(b) Not SHM. The acceleration points away from \(x = 0\). A small displacement grows, and the particle runs away, as from an unstable equilibrium.
(c) Not SHM. The acceleration does point to \(x = 0\), so the particle oscillates. But it is not proportional to \(x\): doubling \(x\) makes it eight times larger.
(d) SHM. Write \(a = -9(x - 2)\). This is \(-\omega^2(x - x_0)\) with \(\omega = 3\,\text{rad/s}\) and the mean position at \(x_0 = 2\,\text{m}\).
Which statement defines simple harmonic motion?
The mean position of a particle in SHM is a position of equilibrium. What kind of equilibrium is it?
A particle performs SHM with amplitude \(A\). What distance does it travel in one complete period?
Return to the corridor clock. The minute hand goes round, so its motion is periodic but not an oscillation. The pendulum goes to and fro about its lowest point, a position of stable equilibrium, so it oscillates. For small swings its acceleration along the arc is very nearly \(-\omega^2\) times its displacement. So the pendulum is in SHM, and the next chapter shows why.
The atoms of a solid, such as a copper wire, sit at positions of stable equilibrium set by the pull and push of their neighbours. They never sit still. Each atom vibrates about its mean position, with a frequency of roughly \(10^{13}\) cycles per second. For small vibrations the motion is very nearly simple harmonic. Heat the wire, and the vibrations grow larger. This picture explains why solids expand when heated, and why they melt when the vibrations grow too large.
- Calling every repeating motion an oscillation.An oscillation must go to and fro about a mean position. Motion round a circle is periodic but not an oscillation.
- Thinking the particle stops at the mean position.It stops for an instant at the extremes. At the mean position it moves fastest.
- Thinking the acceleration is zero at the extremes because the particle stops there.In SHM the acceleration is largest at the extremes, \(\omega^2 A\). A body can be at rest for an instant and still accelerate.
- Measuring \(x\) from one end instead of from the mean position.In \(a = -\omega^2 x\), \(x\) is the displacement from the mean position. If the mean position is at \(x_0\), use \(a = -\omega^2(x - x_0)\).