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DP IB Physics: SL

Revision Notes

Home / IB / Physics: SL / DP / Revision Notes / 4. Waves / 4.1 Oscillations / 4.1.3 SHM Graphs


4.1.3 SHM Graphs


SHM Graphs

Displacement & Velocity

  • The displacement-time graph for an object moving with SHM is a sinusoidal curve if:
    • The object starts to oscillate from the equilibrium position.
    • The equilibrium position is  x = 0 at t = 0
  • The displacement-time graph is a cosine curve if 
    • The object starts to oscillate from the position of maximum displacement. 
    • Maximum displacement is x = x0 at t = 0
  • The maxima and minima on the graph are the values of maximum displacement (x0) of the oscillating object on either side of the equilibrium position 

 

  • The velocity-time graph is obtained by taking the gradients of tangents to all points on the displacement-time graph 

  • The velocity-time graph is a cosine curve if:
    • The object starts to oscillate from the equilibrium position when x = 0 at t = 0 
    • The displacement-time graph is a sine curve
  • The velocity-time graph is a sine curve if:
    • The object starts to oscillate from the position of maximum displacement when x = x0 at t = 0 
    • The displacement-time graph is a cosine curve
  • The maxima and minima on the graph are the values of maximum velocity (v0) of the oscillating object as it passes the equilibrium position
    • The difference in the sign of the velocity accounts for the different directions of the velocity vector as the object passes through the equilibrium position (i.e. from right to left or vice versa)

4-1-3-graphs-of-shm-1

Displacement-time & velocity-time graphs for an object oscillating with SHM. The object starts oscillating from the equilibrium position (x = 0 and t = 0)

Acceleration

  • The acceleration-time graph is obtained by taking the gradients of tangents to all points on the velocity-time graph 
  • The graph is a negative sine curve if:
    • The object starts to oscillate from the equilibrium position when x = 0 at t = 0
  • The graph is a negative cosine curve if:
    • The object starts to oscillate from the position of maximum displacement when x = x0 at t = 0
  • The maxima and minima on the graph are the values of maximum acceleration (a0) of the oscillating object at the positions of maximum displacement (x = x0)
    • Once again, the difference in sign indicates a difference in the direction of the acceleration vector

4-1-3-graphs-of-shm-2 
An acceleration-time graph for an object oscillating with SHM. The object starts oscillating from the equilibrium position (x = 0 and t = 0)

  • Note that all graphs must have the same period
  • The only two differences between the graphs are:
    • The shift in time - i.e. there is a phase difference of 90° between successive graphs
    • The amplitude of the wave form - i.e. the different amplitudes of the three graphs are the values of maximum displacement x0, maximum velocity v0 and maximum acceleration a0 of the oscillating object

Worked Example

Below is the displacement-time graph for an object oscillating with SHM.WE SHM Question Graph, downloadable IB Physics revision notes(i) Determine the period of the oscillations
(ii) Calculate the frequency of the oscillations
(iii) Mark a point on the graph where the velocity is zero, label this with "v = 0"
(iv) Mark a point on the graph where the velocity is maximum and positive, label this with "v0"
(v) Mark a point on the graph where the acceleration is maximum and positive, label this with "a0"
(vi) Determine the value of the maximum velocity v0

(i) Identify the period T of the oscillating object on the graph 

    • Mark the time between any two identical points on the graph (e.g. two peaks) 

WE SHM Step 1, downloadable IB Physics revision notes

T = 0.20 s

(ii) Calculate the frequency f

Step 1: Write down the relationship between frequency and period

Step 2: Substitute the value of the period you have determined in part (i)

f = 5 Hz

(iii) Identify any position of zero velocity on the displacement-time graph and label this "v = 0"

    • The velocity of an object oscillating with SHM is zero at the positions of maximum displacement x = x0
    • Hence, the velocity is zero at any minima or maxima on the displacement-time graph (e.g. at t = 0.10 s)

WE SHM Step 3, downloadable IB Physics revision notes

(iv) Identify any position of maximum positive velocity on the displacement-time graph and label this "v0"

    • An object oscillating with SHM has its maximum velocity at the equilibrium position (x = 0)
    • Velocity is defined as the rate of change of displacement
    • The velocity is the gradient of the tangent to a point of zero displacement
    • The gradient must be positive (e.g. at t = 0.15 s)

WE SHM Step 4, downloadable IB Physics revision notes

(v) Identify any position of maximum positive acceleration on the displacement-time graph and label this "a0"

    • An object oscillating with SHM has its maximum acceleration at the positions of maximum displacement (x = x0)
    • Acceleration is proportional to negative displacement
    • The acceleration is maximum and positive when the displacement is maximum and negative
    • The acceleration is maximum and positive at any minima on the displacement-time graph (e.g. at t = 0.30 s)

WE SHM Step 5, downloadable IB Physics revision notes

(vi)

Step 1: Draw the tangent to the point of maximum positive velocity identified in Step 4 (i.e. at t = 0.15 s) 

WE SHM Step 6, downloadable IB Physics revision notes

Step 2: Calculate the gradient of the tangent to get the value of the maximum velocity v0 in centimetres per second (cm s–1)

Gradient = 67 cm s–1

v0 = 67 cm s–1



  • 1. Measurement & Uncertainties
    • 1.1 Measurements in Physics
      • 1.1.1 SI Units
        • 1.1.2 Using Scientific Notation
          • 1.1.3 Estimating Physical Quantities
          • 1.2 Uncertainties & Errors
            • 1.2.1 Random & Systematic Errors
              • 1.2.2 Calculating Uncertainties
                • 1.2.3 Determining Uncertainties from Graphs
                • 1.3 Vectors & Scalars
                  • 1.3.1 Vector & Scalar Quantities
                    • 1.3.2 Combining & Resolving Vectors
                      • 1.3.3 Solving Vector Problems
                    • 2. Mechanics
                      • 2.1 Motion
                        • 2.1.1 Distance & Displacement
                          • 2.1.2 Speed & Velocity
                            • 2.1.3 Acceleration
                              • 2.1.4 Graphs Describing Motion
                                • 2.1.5 Uniform Acceleration
                                  • 2.1.6 Acceleration of Free Fall Experiment
                                    • 2.1.7 Projectile Motion
                                      • 2.1.8 Terminal Speed
                                      • 2.2 Forces
                                        • 2.2.1 Free-Body Diagrams
                                          • 2.2.2 Newton’s First Law
                                            • 2.2.3 Newton’s Second Law
                                              • 2.2.4 Newton’s Third Law
                                                • 2.2.5 Applying Newton’s Laws of Motion
                                                  • 2.2.6 Friction
                                                  • 2.3 Work, Energy & Power
                                                    • 2.3.1 Kinetic Energy
                                                      • 2.3.2 Gravitational Potential Energy
                                                        • 2.3.3 Elastic Potential Energy
                                                          • 2.3.4 Work Done
                                                            • 2.3.5 Power
                                                              • 2.3.6 Principle of Conservation of Energy
                                                                • 2.3.7 Efficiency
                                                                • 2.4 Momentum & Impulse
                                                                  • 2.4.1 Force & Momentum
                                                                    • 2.4.2 Impulse
                                                                      • 2.4.3 Conservation of Linear Momentum
                                                                        • 2.4.4 Collisions & Explosions
                                                                      • 3. Thermal Physics
                                                                        • 3.1 Thermal Concepts
                                                                          • 3.1.1 Solids, Liquids & Gases
                                                                            • 3.1.2 Temperature
                                                                              • 3.1.3 Internal Energy
                                                                                • 3.1.4 Specific Heat Capacity
                                                                                  • 3.1.5 Specific Latent Heat
                                                                                    • 3.1.6 Phase Change
                                                                                      • 3.1.7 Investigating Thermal Energy
                                                                                      • 3.2 Modelling a Gas
                                                                                        • 3.2.1 Ideal Gas Laws
                                                                                          • 3.2.2 Ideal Gas Equation
                                                                                            • 3.2.3 Kinetic Model of an Ideal Gas
                                                                                              • 3.2.4 Mole Calculations
                                                                                                • 3.2.5 Investigating Gas Laws
                                                                                              • 4. Waves
                                                                                                • 4.1 Oscillations
                                                                                                  • 4.1.1 Properties of Oscillations
                                                                                                    • 4.1.2 Simple Harmonic Oscillations
                                                                                                      • 4.1.3 SHM Graphs
                                                                                                        • 4.1.4 Energy in SHM
                                                                                                        • 4.2 Travelling Waves
                                                                                                          • 4.2.1 Properties of Waves
                                                                                                            • 4.2.2 Transverse & Longitudinal Waves
                                                                                                              • 4.2.3 Electromagnetic Waves
                                                                                                                • 4.2.4 Sound Waves
                                                                                                                • 4.3 Wave Characteristics
                                                                                                                  • 4.3.1 Wavefronts
                                                                                                                    • 4.3.2 Amplitude & Intensity
                                                                                                                      • 4.3.3 Superposition
                                                                                                                        • 4.3.4 Polarisation
                                                                                                                          • 4.3.5 Malus’s Law
                                                                                                                          • 4.4 Wave Behaviour
                                                                                                                            • 4.4.1 Reflection, Refraction & Transmission
                                                                                                                              • 4.4.2 Reflection
                                                                                                                                • 4.4.3 Refraction
                                                                                                                                  • 4.4.4 Determining Refractive Index
                                                                                                                                    • 4.4.5 Single-Slit Diffraction
                                                                                                                                      • 4.4.6 Interference & Path Difference
                                                                                                                                        • 4.4.7 Double-Slit Equation
                                                                                                                                        • 4.5 Standing Waves
                                                                                                                                          • 4.5.1 The Nature of Standing Waves
                                                                                                                                            • 4.5.2 Nodes & Antinodes
                                                                                                                                              • 4.5.3 Boundary Conditions for Standing Waves
                                                                                                                                                • 4.5.4 Harmonics
                                                                                                                                              • 5. Electricity & Magnetism
                                                                                                                                                • 5.1 Electric Fields
                                                                                                                                                  • 5.1.1 Charge & Current
                                                                                                                                                    • 5.1.2 Drift Speed
                                                                                                                                                      • 5.1.3 Potential Difference & DC
                                                                                                                                                        • 5.1.4 Electric Fields
                                                                                                                                                          • 5.1.5 Coulomb’s Law
                                                                                                                                                          • 5.2 Heating Effect of Electric Currents
                                                                                                                                                            • 5.2.1 Circuits
                                                                                                                                                              • 5.2.2 Resistance & Resistors
                                                                                                                                                                • 5.2.3 Kirchhoff’s Circuit Laws
                                                                                                                                                                  • 5.2.4 I-V Characteristics
                                                                                                                                                                    • 5.2.5 Resistivity
                                                                                                                                                                      • 5.2.6 Investigating Resistivity
                                                                                                                                                                        • 5.2.7 Series & Parallel Circuits
                                                                                                                                                                          • 5.2.8 Heating Effect of Current
                                                                                                                                                                            • 5.2.9 Potential Divider Circuits
                                                                                                                                                                            • 5.3 Electric Cells
                                                                                                                                                                              • 5.3.1 Primary & Secondary Cells
                                                                                                                                                                                • 5.3.2 Investigating Electric Cells
                                                                                                                                                                                  • 5.3.3 Terminal Potential Difference
                                                                                                                                                                                    • 5.3.4 Electromotive Force & Internal Resistance
                                                                                                                                                                                    • 5.4 Magnetic Effects of Electric Currents
                                                                                                                                                                                      • 5.4.1 Magnetic Fields
                                                                                                                                                                                        • 5.4.2 Magnetic Force on a Charge
                                                                                                                                                                                          • 5.4.3 Magnetic Force on a Current-Carrying Conductor
                                                                                                                                                                                            • 5.4.4 Solving Problems Involving Magnetic Forces
                                                                                                                                                                                          • 6. Circular Motion & Gravitation
                                                                                                                                                                                            • 6.1 Circular Motion
                                                                                                                                                                                              • 6.1.1 Circular Motion
                                                                                                                                                                                                • 6.1.2 Centripetal Force
                                                                                                                                                                                                  • 6.1.3 Centripetal Acceleration
                                                                                                                                                                                                    • 6.1.4 Applications of Circular Motion
                                                                                                                                                                                                    • 6.2 Newton’s Law of Gravitation
                                                                                                                                                                                                      • 6.2.1 Newton’s Law of Gravitation
                                                                                                                                                                                                        • 6.2.2 Circular Orbits
                                                                                                                                                                                                          • 6.2.3 Gravitational Field Strength
                                                                                                                                                                                                        • 7. Atomic, Nuclear & Particle Physics
                                                                                                                                                                                                          • 7.1 Discrete Energy & Radioactivity
                                                                                                                                                                                                            • 7.1.1 Discrete Energy Levels
                                                                                                                                                                                                              • 7.1.2 Calculating Discrete Energies
                                                                                                                                                                                                                • 7.1.3 Emission & Absorption Spectrum
                                                                                                                                                                                                                  • 7.1.4 Isotopes & Radioactive Decay
                                                                                                                                                                                                                    • 7.1.5 Background Radiation
                                                                                                                                                                                                                      • 7.1.6 Alpha, Beta & Gamma Particles
                                                                                                                                                                                                                        • 7.1.7 Decay Equations
                                                                                                                                                                                                                          • 7.1.8 Half-Life
                                                                                                                                                                                                                            • 7.1.9 Investigating Half-Life
                                                                                                                                                                                                                            • 7.2 Nuclear Reactions
                                                                                                                                                                                                                              • 7.2.1 Atomic Mass Unit
                                                                                                                                                                                                                                • 7.2.2 Mass Defect & Nuclear Binding Energy
                                                                                                                                                                                                                                  • 7.2.3 Nuclear Fission & Fusion
                                                                                                                                                                                                                                    • 7.2.4 Binding Energy per Nucleon Curve
                                                                                                                                                                                                                                    • 7.3 The Structure of Matter
                                                                                                                                                                                                                                      • 7.3.1 Discovery of the Nucleus
                                                                                                                                                                                                                                        • 7.3.2 Quarks & Leptons
                                                                                                                                                                                                                                          • 7.3.3 Hadrons, Baryons & Mesons
                                                                                                                                                                                                                                            • 7.3.4 Particle Conservation Laws
                                                                                                                                                                                                                                              • 7.3.5 Fundamental Forces
                                                                                                                                                                                                                                                • 7.3.6 Feynman Diagrams & Confinement
                                                                                                                                                                                                                                                  • 7.3.7 The Standard Model of Particle Physics
                                                                                                                                                                                                                                                • 8. Energy Production
                                                                                                                                                                                                                                                  • 8.1 Energy Sources
                                                                                                                                                                                                                                                    • 8.1.1 Specific Energy
                                                                                                                                                                                                                                                      • 8.1.2 Sankey Diagrams
                                                                                                                                                                                                                                                        • 8.1.3 Primary & Secondary Energy Sources
                                                                                                                                                                                                                                                          • 8.1.4 Energy Resources
                                                                                                                                                                                                                                                            • 8.1.5 Energy Generation
                                                                                                                                                                                                                                                              • 8.1.6 Safety Issues in Nuclear Power
                                                                                                                                                                                                                                                              • 8.2 Thermal Energy Transfer
                                                                                                                                                                                                                                                                • 8.2.1 Conduction, Convection & Thermal Radiation
                                                                                                                                                                                                                                                                  • 8.2.2 Black-Body Radiation
                                                                                                                                                                                                                                                                    • 8.2.3 The Solar Constant, Albedo & Emissivity
                                                                                                                                                                                                                                                                      • 8.2.4 The Greenhouse Effect
                                                                                                                                                                                                                                                                        • 8.2.5 Earth's Surface–Atmosphere System
                                                                                                                                                                                                                                                                      Ashika Vijay

                                                                                                                                                                                                                                                                      Author: Ashika

                                                                                                                                                                                                                                                                      Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.


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