7.1.1

Simple Harmonic Motion

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Simple Harmonic Motion (SHM)

SHM is any motion in which the acceleration is directed towards a fixed point (or equilibrium position for 1-D examples) AND is directly proportional to the negative of the displacement.

Formulae

Formulae

  • Mathematically, the definition of SHM can be expressed in a formula:
    • a=−ω2xa=-\omega^2 x
      • a is the acceleration.
      • ω=2πf\omega = 2\pi f, where f is the frequency of oscillation
      • x is the displacement.
Acceleration-displacement graph

Acceleration-displacement graph

  • The acceleration-displacement graph is a straight-line graph that passes through the origin.
  • The angular frequency can be found from the gradient:
    • gradient =−ω2= -\omega^2
    • ω=−gradient\omega = \sqrt{-gradient}
Calculating maximum acceleration

Calculating maximum acceleration

  • If we are given the time period, T (or the frequency f), ω² can be calculated.
  • The maximum acceleration is given by:
    • amax=ω2Aa_{max}=\omega^2A
      • where A is the maximum displacement.

Simple Harmonic Motion (SHM) Graphs

SHM can be represented graphically.

Acceleration and displacement

Acceleration and displacement

  • The acceleration can be found graphically, and so the displacement or vice versa from a graph.
    • E.g. if you have a displacement-time graph, you can calculate the period and angular frequency.
  • The displacement at any point can be read off the graph. This means you can calculate the acceleration (and then even the net force) because a=−ω2xa=-\omega^2 x.
Velocity and displacement

Velocity and displacement

  • The velocity at any given time is found from the gradient of a displacement-time graph.
  • By finding the gradient at each point in time, we can produce a velocity-time graph from a displacement-time graph.
Velocity and acceleration

Velocity and acceleration

  • In a similar way, you can use a velocity-time graph to get the corresponding acceleration-time graph. This is because the gradient at a point on any velocity-time graph gives the acceleration at that moment in time.
  • You can then see that the acceleration-time graph is exactly the same shape as the displacement-time graph, reflected in the x-axis.

Simple Harmonic Equations

The condition for simple harmonic motion is that the acceleration is directed towards a fixed point and that the magnitude of the acceleration is proportional to the negative of the displacement.

Oscillations

Oscillations

  • For SHM the acceleration must be proportional to the negative of the displacement. aα−xa{\alpha}-x
  • If an object is under SHM it's displacement is found by x=x0cos⁡(ωt)x=x_0\cos({\omega}t) where x_0 is the maximum displacement.
  • The defining equation of SHM is a=−ω2xa=-{\omega}^2x
Velocity

Velocity

  • The velocity of an object in simple harmonic motion is a sine function:
    • v=v0sin⁡(ωt)v=v_0\sin(\omega t)
  • Where v0v_0 is the maximum velocity.
  • We can also calculate the velocity of an object in SHM by using the equation:
    • v=±ω(x02−x2)v = \pm\omega(\sqrt{x_0^2-x^2})

Maximum points

  • The maximum displacement can be found when cos⁡(ωt)=1\cos({\omega}t)=1
  • The maximum displacement x0x_0 is the amplitude of the cosine wave.
  • The maximum speed is given by v0=ωx0v_0={\omega}x_0
  • The maximum acceleration is given by a=ω2x0a={\omega}^2x_0
Jump to other topics
1

Physical Quantities & Units

1.1

Physical Quantities & Units

1.1.1Use of SI Units1.1.2SI Prefixes, Standard Form & Converting Units1.1.3End of Topic Test - Units & Prefixes1.1.4Avogadro's Constant1.1.5Scalars & Vectors
1.2

Errors & Uncertainty

1.2.1Measuring Length, Time & Volume1.2.2Measuring Temperature1.2.3Circuit Measurements1.2.4Limitation of Physical Measurements1.2.5Uncertainty1.2.6Estimation1.2.7Displaying Data1.2.8End of Topic Test - Measurements & Errors
2

Kinematics

2.1

Equations of Motion

2.1.1Motion in a Straight Line2.1.2Graphs of Motion2.1.3Bouncing Ball Example2.1.4End of Topic Test - Motion in a Straight Line2.1.5Acceleration Due to Gravity
3

Dynamics

3.1

Momentum & Newton's Laws of Motion

3.1.1Mass & Inertia3.1.2Newton's Laws3.1.3Momentum3.1.4End of Topic Test - Newton's Laws & Momentum3.1.5A-A* (AO3/4) - Newton's Third Law
3.2

Non-Uniform Motion

3.2.1Projectile Motion
3.3

Linear Momentum & Conservation

3.3.1Conservation of Momentum
4

Force, Density & Pressure

4.1

Force, Density & Pressure

4.1.1Fields4.1.2Force in Uniform Fields4.1.3Friction4.1.4Buoyancy4.1.5Terminal Speed4.1.6End of Topic Test - Acceleration Due to Gravity4.1.7Centre of Mass4.1.8Forces & Equilibrium4.1.9End of Topic Test - Scalars & Vectors4.1.10Moments4.1.11End of Topic Test - Moments & Centre of Mass4.1.12Density4.1.13Pressure
5

Work, Energy & Power

5.1

Work, Energy & Power

5.1.1Work & Energy5.1.2Power & Efficiency5.1.3Conservation of Energy5.1.4End of Topic Test - Work, Energy & Power
5.2

Gravitational Fields

5.2.1Newton's Law5.2.2Gravitational Field Strength5.2.3Gravitational Potential
6

Deformation of Solids

6.1

Materials

6.1.1Bulk Properties of Solids6.1.2Energy in Materials6.1.3Young Modulus6.1.4End of Topic Test - Materials
7

Waves

7.1

Simple Harmonic Motion

7.1.1Simple Harmonic Motion7.1.2Simple Harmonic Systems7.1.3Energy in Simple Harmonic Motion7.1.4Resonance7.1.5End of Topic Test - Simple Harmonic Motion7.1.6A-A* (AO3/4) - Simple Harmonic Motion7.1.7End of Topic Test - Gravitational Fields
7.2

Waves

7.2.1Progressive Waves7.2.2Intensity of Waves7.2.3Wave Speed & Phase Difference7.2.4Longitudinal & Transverse Waves7.2.5End of Topic Test - Progressive Waves7.2.6Electromagnetic Waves7.2.7Doppler Effect7.2.8Sound Waves7.2.9Measuring Sound Waves7.2.10End of Topic Test - Waves7.2.11Ultrasound Imaging7.2.12Ultrasound Imaging 2
8

Superposition

8.1

Superposition

8.1.1Stationary Waves8.1.2Stationary Waves 28.1.3End of Topic Test - Stationary Waves8.1.4A-A* (AO3/4) - Stationary Waves8.1.5Interference8.1.6Interference 28.1.7End of Topic Test - Interference8.1.8Diffraction8.1.9Diffraction Gratings8.1.10End of Topic Test - Diffraction
9

Thermal Physics

9.1

Circular Motion

9.1.1Circular Motion9.1.2Circular Motion 29.1.3End of Topic Test - Circular Motion
9.2

Thermal Physics

9.2.1Temperature9.2.2Measuring Temperature9.2.3Ideal Gas Law9.2.4Ideal Gases9.2.5Boyle's Law & Charles' Law9.2.6Molecular Kinetic Theory Model9.2.7Molecular Kinetic Theory Model 29.2.8Thermal Energy Transfer9.2.9Thermal Energy Transfer Experiments9.2.10End of Topic Test - Thermal Energy & Ideal Gases9.2.11First Law of Thermodynamics
10

Communication

10.1

Communication Channels

10.1.1Communication Channels10.1.2Modulation10.1.3Attenuation
10.2

Digital Communication

10.2.1Digital & Analogue Signals10.2.2Representing Sound
11

Electric Fields

11.1

Electric Fields

11.1.1Coulomb's Law11.1.2Electric Field Strength11.1.3Electric Field Strength 211.1.4Electric Potential11.1.5End of Topic Test - Electric Fields11.1.6A-A* (AO3/4) - Electric and Gravitational Field
11.2

Capacitance

11.2.1Capacitance11.2.2Parallel Plate Capacitor11.2.3Energy Stored by a Capacitor11.2.4Capacitor Discharge11.2.5Capacitor Charge
12

Current Electricity

12.1

Current Electricity

12.1.1Basics of Electricity12.1.2Mean Drift Velocity12.1.3Current-Voltage Characteristics12.1.4End of Topic Test - Basics of Electricity12.1.5Resistivity12.1.6End of Topic Test - Resistivity & Superconductors12.1.7Power and Conservation12.1.8Microphones12.1.9Components12.1.10Relays12.1.11Strain Gauges
12.2

DC Circuits

12.2.1Circuits12.2.2Potential Divider12.2.3Emf & Internal Resistance12.2.4End of Topic Test - Power & Potential
13

Magnetic Fields

13.1

Magnetic Fields

13.1.1Magnetic Flux Density13.1.2End of Topic Test - Capacitance & Flux Density13.1.3Moving Charges in a Magnetic Field13.1.4Magnetic Flux & Flux Linkage13.1.5Electromagnetic Induction13.1.6Electromagnetic Induction 213.1.7Magnetic Flux Density13.1.8Magnetic Resonance Scanning
13.2

Alternating Currents

13.2.1Alternating Currents13.2.2Operation of a Transformer13.2.3End of Topic Test - Electromagnetic Induction & AC13.2.4Rectification
14

Modern Physics

14.1

Quantum Physics

14.1.1The Photoelectric Effect14.1.2The Photoelectric Effect Explanation14.1.3End of Topic Test - The Photoelectric Effect14.1.4Collisions of Electrons with Atoms14.1.5Energy Levels & Photon Emission14.1.6Wave-Particle Duality14.1.7End of Topic Test - Absorption & Emission14.1.8Band Theory14.1.9Diagnostic X-Rays14.1.10X-Ray Image Processing14.1.11Absorption of X-Rays14.1.12CT Scanners
14.2

Nuclear Physics

14.2.1Rutherford Scattering14.2.2Atomic Model14.2.3Isotopes14.2.4Stable & Unstable Nuclei14.2.5A-A* (AO3/4) - Stable & Unstable Nuclei14.2.6Alpha & Beta Radiation14.2.7Gamma Radiation14.2.8Particles, Antiparticles & Photons14.2.9Quarks & Antiquarks14.2.10Particle Interactions14.2.11Radioactive Decay14.2.12Half Life14.2.13End of Topic Test - Radioactivity14.2.14Nuclear Instability14.2.15Mass & Energy14.2.16Binding Energy14.2.17A-A* (AO3/4) - Nuclear Fusion

Practice questions on Simple Harmonic Motion

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

  1. 1
    Which statements about simple harmonic motion are true?True / false
  2. 2
    What is the correct equation for the maximum acceleration? Multiple choice
  3. 3
    What can be found using the gradient of an acceleration-displacement graph?Multiple choice
  4. 4
    Which statements about SHM graphs are true?True / false
  5. 5
    How do an acceleration-time and a displacement-time graph differ in simple harmonic motion?Multiple choice
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End of Topic Test - Materials

Simple Harmonic Systems