doPhysics · Oscillations

Kinematics of SHM

Why this chapter

Why this chapter exists

A swing, a guitar string, the air in a flute and the atoms of a solid all move to and fro. Almost every oscillation in nature, when it is small, is one particular motion: simple harmonic motion. This chapter describes that motion completely, without asking what causes it. It starts from one law: the acceleration is \(-\omega^2\) times the displacement. From that law it finds where the particle is and how fast it moves at every instant, and how long each stretch takes. The next chapter, Dynamics of SHM, finds the forces behind it.

You will learn

By the end of this chapter you will

Use radians and draw the graphs of sine and cosine. Shift them by a phase, and write \(a\sin\theta + b\cos\theta\) as one sine.
Differentiate \(\sin\omega t\) and \(\cos\omega t\) with the chain rule.
Tell periodic motion, oscillation and SHM apart, and define SHM by \(a = -\omega^2x\).
Name and use the amplitude, period, frequency, angular frequency, phase and initial phase of an SHM.
Derive \(x = A\sin(\omega t + \phi)\) with its velocity and acceleration, and show that the cosine form is the same motion.
Use \(v = \pm\omega\sqrt{A^2 - x^2}\) and read the a-x line and the v-x ellipse.
Use the reference circle and phasors to find phase differences and the time between two positions.

Before you begin

From Kinematics of Circular Motion you should know angular velocity, the radian, uniform circular motion and the centripetal acceleration \(\omega^2r\). From Motion in One Dimension you should know velocity and acceleration as derivatives. From Potential Energy and Conservation of Energy you should know stable equilibrium. You should know the sine and cosine of the standard angles.

Already know this?

Lessons

The Mathematics of Oscillation

The tools that every later lesson uses. Radians, the graphs of sine and cosine, what a phase added to the angle does, why a sine plus a cosine is one sine, and the derivatives of \(\sin\omega t\) and \(\cos\omega t\).

Radians and the graphs of sine and cosineShifting a sine by a phaseA sine plus a cosine is one sineDerivatives of sine and cosine

13 min0%

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Periodic Motion, Oscillation and SHM

Three nested ideas. A periodic motion repeats after a fixed time; an oscillation is a periodic motion to and fro about a mean position; simple harmonic motion is the oscillation whose acceleration is \(-\omega^2\) times the displacement.

Periodic motion repeats after a fixed timeOscillation is to and fro about a mean positionShm is defined by a equals minus omega squared x

14 min0%

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Amplitude, Period, Frequency and Phase

The names of the parts of \(x = A\sin(\omega t + \phi)\). The amplitude sets the size, the period and the frequency set the pace, the phase fixes the state of motion at each instant, and the initial phase fixes where the motion starts.

Amplitude period and frequencyPhase and angular frequencyInitial phase sets the start

13 min0%

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The Equations of SHM

From the law \(a = -\omega^2x\) to the motion itself. The a-x line, the solution \(x = A\sin(\omega t + \phi)\) and why \(\cos\) is the same motion, velocity and acceleration against time, and velocity against position, whose graph is an ellipse.

The a x graph is a line of slope minus omega sqX equals a sin omega t plus phi solves shmVelocity and acceleration against timeVelocity against position is an ellipse

15 min0%

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SHM as the Shadow of Circular Motion

A point going round a circle at a steady rate casts a shadow on a diameter, and the shadow moves in SHM. The reference circle, the phasor and phase difference, and the time between two positions read as an angle swept.

Shm is the shadow of uniform circular motionThe phasor and phase differenceTime between two positions on the reference circle

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

amplitude: The greatest displacement of an oscillating body from its mean position, \(A\). It is always positive.
angular frequency: The constant \(\omega\) of an SHM, in rad/s. It is the rate at which the phase grows, and \(\omega = 2\pi f = 2\pi/T\).
antiphase: The relation of two SHMs whose phase difference is \(\pi\). Their displacements are always opposite in sign.
chain rule: The rule for differentiating a function of a function. The rates multiply: \(d(\sin\omega t)/dt = \omega\cos\omega t\).
differential equation: An equation that contains a derivative of an unknown function, such as \(d^2x/dt^2 = -\omega^2 x\). Its solution is a function, not a number.
displacement: In SHM, the signed distance \(x\) of the particle from the mean position, measured along the line of motion.
ellipse: A closed oval curve with the equation \(x^2/a^2 + y^2/b^2 = 1\), of half-width \(a\) and half-height \(b\). The velocity-position graph of SHM is an ellipse.
extreme position: One of the two points where an oscillating body stops for an instant and turns back. In SHM they are at \(x = \pm A\).
frequency: The number of complete oscillations in one second, \(f = 1/T\), measured in hertz.
hertz: The unit of frequency, one oscillation per second, \(1\,\text{Hz} = 1\,\text{s}^{-1}\).
initial phase: The phase of an SHM at \(t = 0\), \(\varphi\), also called the phase constant. It fixes where the motion starts and which way it first goes.
isochronous: Having a period that does not depend on the amplitude. Every SHM is isochronous, because \(T = 2\pi/\omega\) contains no \(A\).
lead: To be ahead in phase. Of two SHMs, the one with the larger phase leads and reaches each peak earlier, by \(\Delta\varphi/\omega\).
mean position: The fixed point about which a body oscillates. It is a position of stable equilibrium, where the acceleration of an SHM is zero.
oscillation: A periodic motion to and fro about a fixed mean position, between two extreme positions. Also called a vibration.
period: The smallest time after which a periodic motion repeats itself, \(T\). In SHM, \(T = 2\pi/\omega\). For a function of an angle, such as \(\sin\theta\), the period is \(2\pi\).
periodic motion: A motion that repeats itself exactly after a fixed interval of time, so that \(x(t + T) = x(t)\) at every instant.
phase: The angle \(\Phi = \omega t + \varphi\) inside the sine of an SHM. It fixes both the displacement and the direction of motion at an instant.
phase difference: The difference between the phases of two SHMs of the same angular frequency, \(\Delta\varphi\). It stays constant, and it is the angle between their phasors.
phasor: A rotating arrow of length \(A\) that turns at the angular frequency \(\omega\). Its angle is the phase, and its projection on the vertical axis is the displacement of the SHM.
radian: The angle at the centre of a circle whose arc equals the radius. A full turn is \(2\pi\) rad, so \(\pi\) rad \(= 180°\).
reference circle: The circle of radius \(A\) round which a point moves at a steady angular velocity \(\omega\). The shadow of the point on a diameter performs the SHM \(x = A\sin(\omega t + \varphi)\).
simple harmonic motion: An oscillation in which the acceleration is proportional to the displacement from the mean position and directed towards it, \(a = -\omega^2 x\). SHM for short.
sinusoid: A curve with the shape of the graph of \(\sin\theta\), whether written with a sine or a cosine. It repeats every \(2\pi\) of its angle.
stable equilibrium: A position where the net force on a body is zero, and a small displacement produces a force back towards it. A body displaced from it oscillates.