Oscillation is one of the most common forms of motion in nature. A pendulum swings, a spring bounces, a guitar string vibrates, and the molecules of a solid vibrate about their mean positions. In each case, the body moves back and forth about a mean or equilibrium position. This chapter studies the simplest and most important type of oscillation - simple harmonic motion (SHM).
Simple harmonic motion is the motion in which the acceleration of the body is proportional to its displacement from the equilibrium position and is always directed toward the equilibrium position. Many real oscillations are approximately simple harmonic for small amplitudes, which makes SHM an invaluable model.
We study the displacement, velocity, and acceleration of a particle in SHM, its energy changes, and its period for a spring-mass system and a simple pendulum. We also consider damped oscillations, forced oscillations, and resonance, which explain phenomena as varied as the swinging of a clock pendulum and the collapse of structures under resonant wind forces.
A motion that repeats itself at regular intervals of time is called periodic motion. The time after which the motion repeats is the period T, and the number of oscillations per unit time is the frequency:
f = 1 / T
The SI unit of frequency is the hertz (Hz), where 1 Hz = 1 oscillation per second. Oscillatory motion is a special type of periodic motion in which a body moves back and forth about a mean position. Not all periodic motions are oscillatory - for example, the uniform rotation of the Earth is periodic but not oscillatory.
In oscillatory motion, the body returns to its mean position because of a restoring force that always acts toward the mean position. For small displacements, the restoring force in many systems is proportional to the displacement, which leads to simple harmonic motion.
A particle performs simple harmonic motion if its acceleration is directly proportional to its displacement from the equilibrium position and is directed toward that position:
a = - omega^2 x
where omega is the angular frequency. The negative sign shows that the acceleration is opposite to the displacement. The displacement of a particle in SHM varies sinusoidally with time:
x = A sin(omega t + phi)
where A is the amplitude, the maximum displacement from the mean position, and phi is the initial phase. The angular frequency is related to the period by:
omega = 2 pi / T = 2 pi f
The velocity of the particle is obtained by differentiating the displacement:
v = dx/dt = omega * A * cos(omega t + phi) = omega * sqrt(A^2 - x^2)
The velocity is maximum at the mean position (x = 0) and zero at the extreme positions (x = +/- A). The acceleration is:
a = - omega^2 x
so the acceleration is zero at the mean position and maximum at the extreme positions.
The argument (omega t + phi) in the displacement equation is called the phase of the motion. It determines the state of the particle at any time. The initial phase phi determines the displacement at t = 0. Two particles in SHM are in phase if their phase difference is an integer multiple of 2 pi, and in opposite phase if the phase difference is an odd multiple of pi.
Simple harmonic motion has a beautiful geometric interpretation: it is the projection of uniform circular motion along a diameter of the circle. If a particle moves uniformly around a circle of radius A with angular speed omega, its projection on a diameter moves with displacement x = A sin(omega t + phi). This relation connects the kinematics of SHM to the uniform circular motion studied earlier.
This geometric picture explains why the velocity and acceleration have the forms they do, and why the maximum speed is omega A and the maximum acceleration is omega^2 A.
A particle executing SHM has both kinetic and potential energy. The kinetic energy is maximum at the mean position where the speed is maximum, and zero at the extremes. The potential energy is maximum at the extremes and zero at the mean position. For a particle of mass m oscillating with angular frequency omega and amplitude A:
K = (1/2) m omega^2 (A^2 - x^2)
U = (1/2) m omega^2 x^2
The total mechanical energy is the sum:
E = K + U = (1/2) m omega^2 A^2
The total energy is constant, as required by the conservation of mechanical energy. It is proportional to the square of the amplitude and the square of the angular frequency. The kinetic and potential energies oscillate at twice the frequency of the displacement, and their average values over one period are equal, each being half the total energy.
The most basic example of SHM is a mass attached to a spring. When a mass m is attached to a spring of force constant k and displaced by x, the restoring force is:
F = - k x
By Newton's second law, m a = -k x, which gives a = -(k/m)x. Comparing with a = -omega^2 x, we find the angular frequency and period:
omega = sqrt(k / m) and T = 2 pi sqrt(m / k)
The period of a spring-mass system depends only on the mass and the spring constant, and is independent of the amplitude and of g. If two springs are attached to a mass in series, the effective constant is given by 1/k_eff = 1/k1 + 1/k2, while in parallel the constants add: k_eff = k1 + k2.
A simple pendulum consists of a point mass (the bob) suspended by a massless, inextensible string from a fixed support. When the bob is displaced through a small angle and released, it oscillates. The restoring force is the component of the weight along the tangent, F = -m g sin theta. For small angles, sin theta is approximately equal to theta (measured in radians), and the motion is simple harmonic with:
T = 2 pi sqrt(L / g)
where L is the length of the pendulum. The period of a simple pendulum depends only on its length and the acceleration due to gravity; it is independent of the mass of the bob and of the amplitude (for small amplitudes). This is why a pendulum clock keeps time. Since g varies with location, the period of a pendulum clock changes with altitude and latitude.
In real systems, the amplitude of oscillation decreases gradually because of damping forces such as air resistance and internal friction. Such motion is called damped oscillation. The damping force is proportional to the velocity and opposite to it. The mechanical energy of a damped oscillator decreases with time, being converted into heat. A system is underdamped if it oscillates with decreasing amplitude, critically damped if it returns to equilibrium without oscillating in the shortest time, and overdamped if it returns slowly without oscillating.
Forced oscillations occur when a periodic external force acts on a system. The system then oscillates at the frequency of the driving force. When the driving frequency equals the natural frequency of the system, the amplitude becomes very large; this is called resonance. Resonance explains why soldiers break step while crossing a bridge, why radio tuning works, and why a swing can be pushed to a large amplitude with small periodic pushes. In musical instruments and the human ear, resonance is fundamental to producing and detecting sound.
| Quantity | Formula in SHM |
|---|---|
| Displacement | x = A sin(omega t + phi) |
| Velocity | v = omega sqrt(A^2 - x^2) |
| Acceleration | a = - omega^2 x |
| Kinetic energy | K = (1/2) m omega^2 (A^2 - x^2) |
| Potential energy | U = (1/2) m omega^2 x^2 |
| Total energy | E = (1/2) m omega^2 A^2 |
| System | Period |
|---|---|
| Spring-mass | T = 2 pi sqrt(m/k) |
| Simple pendulum | T = 2 pi sqrt(L/g) |
| Angular frequency | omega = 2 pi / T = 2 pi f |
In this chapter we studied oscillatory motion, focusing on the fundamental case of simple harmonic motion. We learned that SHM is characterised by a = -omega^2 x and x = A sin(omega t + phi), and we derived the velocity and acceleration at every point of the motion. The energy analysis showed that kinetic and potential energy exchange constantly while the total energy E = (1/2) m omega^2 A^2 remains constant. The spring-mass system with period T = 2 pi sqrt(m/k) and the simple pendulum with period T = 2 pi sqrt(L/g) provided concrete examples. Finally, damped, forced, and resonant oscillations connected SHM to the real world. These ideas now extend naturally to the study of waves, where oscillating particles transport energy through a medium.