Physics - Classical Mechanics - Simple Harmonic Motion and Reference Circle

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Introduction

Hey it's a me again @drifter1!

Today we continue with Physics, and more specifically the branch of "Classical Mechanics" in order to cover Simple Harmonic Motion and the Reference Circle.

So, without further ado, let's get straight into it!

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SHM and Smooth Circular Motion

A Smooth Harmonic Motion model can also be constructed from Uniform Circular Motion, as the mathematical equations that describe SHM relate to the concepts of circular motion.

Starting off from a circular motion with a radius A (amplitude) and angular frequency ω, the corresponding SHM is the projection of the so called rotating phase vector on the horizontal axis (or diameter). The projection is thus equal to the cosinus of each individual angle (or phase) φ, times the amplitude A, as shown below.

If at t = 0, the angular position is φ, as it rotates this angle (or phase) changes by ωt, meaning that the angular position at any t is equal to φ + ωt. The projection to the horizontal axis is then equal to:

which is the SHM equation for displacement that we proved last time!

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Reference Circle

The diagram shown in the previous section is the so called Reference Circle. As the point P moves anti-clockwise with an angular frequency of ω, the vector from the center O to this point forms an angle of φ to the horizontal axis. Projecting this rotating phase vector to the horizontal axis yields the position of the corresponding SHM.

Additionally, the horizontal components of the velocity and acceleration of this circular motion give us the velocity and acceleration fo the corresponding SHM. The velocity of point P is equal to ωΑ, whilst its acceleration is ωA², giving us the equations:

that where proven last time.

Some key phase angles and the corresponding displacements are:

Phase (φ)	Displacement (x)
0	A
π / 6 (30°)	~ 0.86A
π / 4 (45°)	~ 0.7A
π / 3 (60°)	0.5A
π / 2 (90°)	0
π (180°)	-A
3π / 2 (270°)	0

Now we are ready for full-on examples, which will come next time!

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RESOURCES:

References

1. https://openstax.org/books/university-physics-volume-1/pages/15-3-comparing-simple-harmonic-motion-and-circular-motion
2. https://courses.lumenlearning.com/physics/chapter/16-6-uniform-circular-motion-and-simple-harmonic-motion/
3. https://demo.webassign.net/ebooks/cj6demo/pc/c10/read/main/c10x10_2.htm?cf_chl_jschl_tk=ogXYI1iTjtuSIcA7e3hgDmdc9lsyC4iBLYKHtC5bwnc-1640697000-0-gaNycGzNCD0

Images

1. 

Mathematical equations used in this article, where made using quicklatex.

Diagrams where made in draw.io.

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Previous articles of the series

Rectlinear motion

• Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
• Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
• Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
• Rectlinear motion exercises -> examples and tasks in rectlinear motion

Plane motion

• Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
• Projectile motion as a plane motion -> missile/bullet motion as a plane motion
• Smooth Circular motion -> smooth circular motion theory
• Plane motion exercises -> examples and tasks in plane motions

Newton's laws and Applications

• Force and Newton's first law -> force, 1st law
• Mass and Newton's second law -> mass, 2nd law
• Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
• Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
• Contact forces and friction -> contact force, friction
• Dynamics of Circular motion -> circular motion dynamics, applications
• Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
• Contact force and friction examples -> exercises in force and friction
• Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
• Advanced Newton law examples -> advanced (more difficult) exercises

Work and Energy

• Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
• Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
• Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
• Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
• Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
• Internal Energy and Work -> Internal Energy, Internal Work

Momentum and Impulse

• Conservation of Momentum -> Momentum, Conservation of Momentum
• Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
• Collision Examples -> Various Elastic and Inelastic Collision Examples
• Impulse -> Impulse with Example
• Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
• Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis

Angular Motion

• Angular motion basics -> Angular position, velocity and acceleration
• Rotation with constant angular acceleration -> Constant angular acceleration, Example
• Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
• Parallel Axis Theorem -> Parallel axis theorem with example
• Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
• Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
• Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
• Angular Momentum -> Angular Momentum and its conservation
• Explaining the Physics behind Mechanical Gyroscopes -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers
• Exercises around Angular motion -> Angular motion examples

Equilibrium and Elasticity

• Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
• Force Couple System -> Force Couple System, Example
• Tensile Stress and Strain -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio
• Volumetric Stress and Strain -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility
• Cross-Sectional Stress and Strain -> Shear Stress, Shear Strain, Shear Modulus
• Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
• Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
• Exercises on Elasticity and Plasticity -> Young Modulus, Bulk Modulus and Shear Modulus Examples

Gravity

• Newton's Law of Gravitation -> Newton's Law of Gravity, Gravitational Constant G
• Weight: The Force of Gravity -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System
• Gravitational Fields -> Gravitational Field Mathematics and Visualization
• Gravitational Potential Energy -> Gravitational Potential Energy, Potential and Escape Velocity
• Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
• Exercises around Newtonian Gravity (part2) -> Examples on Gravitational Fields and Potential Energy
• Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites
• Kepler's Laws of Planetary Motion -> Kepler's Story, Elliptical Orbits, Kepler's Laws
• Spherical Mass Distributions -> Spherical Mass Distribution, Gravity Outside and Within a Spherical Shell, Simple Examples
• Earth's Rotation and its Effect on Gravity -> Gravity on Earth, Apparent Weight
• Black Holes and Schwarzschild Radius -> Black Holes (Creation, Types, How To "See" Them), Schwarzschild Radius

Periodic Motion

• Periodic Motion Fundamentals -> Fundamentals (Period, Frequency, Angular Frequency, Return Force, Acceleration, Velocity, Amplitude), Simple Harmonic Motion, Example
• Energy in Simple Harmonic Motion -> Forms of Energy in SHM (Potential, Kinetic, Total and Maximum Energy, Maximum Velocity), Simple Example
• Simple Harmonic Motion Equations -> SHM Equations (Displacement, Velocity, Acceleration, Phase Angle, Amplitude)

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Final words | Next up

And this is actually it for today's post!

Next time we will get into exercises on the topic of Simple Harmonic Motion...

See ya!

Keep on drifting!