# Physics - Classical Mechanics - Periodic Motion Fundamentals

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## Introduction

Hey it's a me again @drifter1!

After almost a year has passed, it's time to continue with **Physics**, and more specifically the branch of "**Classical Mechanics**" (more than enough of a break!). We have finished the topic of Gravity when we covered Black Holes in the previous article on December 18th, 2020. The next chapter is **Periodic Motion**, which is also known as **Oscillation**. In this post we will start with a quick introduction to the **Fundamentals** of such motion!

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

## Fundamentals

A periodic motion is a motion that **repeats** itself over regular time intervals. One such complete repetion is known as a **cycle**.

### Period and Frequency

The duration of a complete cycle of oscillation is called a **period** and denoted by the letter *T*. The inverse, which is the number of cycles completed in a specific interval of time, is called **frequency**, and equal to the reciprocal of the period:

The period is measured in seconds (*s*), whilst the frequency is measured in Hertz (*Hz*).

### Angular Frequency

In some motions its easier to use something known as **angular frequency**, denoted by the letter *ω*. This frequency refers to angular displacement per time (*rad / s*), and is given by:

### Hooke's Law and Return Force

The simplest oscillation can be expressed using spring deformation, and thus Hooke's Law. Let's say, that the center *O* in the coordinate system of such a mass on a spring system specifies the equilibrium position, and *x* specifies the deformation. Then a force, known as the **return force**, given by:

will try to bring the system back to equilibrium.

### Acceleration and Velocity

From Newton's Second Law, the corresponding **acceleration** is:

The value of the acceleration is of course constantly changing! As the system comes close to the equilibrium position, the acceleration decreases, but the velocity increases. At equilibrium the velocity is maximum (acceleration is zero), whilst at maximum deformation the acceleration is maximum (velocity is zero).

### Amplitude

The maximum displacement (deformation) from the equilibrium position is known as the **amplitude** of oscillation, and commonly denoted by *A*. It's basically equal to the maximum value of *|x|*.

## Simple Harmonic Motion

A periodic motion which is described by Hooke's Law is known as a **Simple Harmonic Motion**.

In such a motion, the period of the mass on a spring is given by the equation:

The corresponding frequency is:

Calculating the velocity, acceleration etc. for a specific amplitude, frequency etc. is more complicated. The equations that describes these quantities are solutions to differential equations, and will be covered later on!

## Example

Let's suppose a system with a spring of constant *k = 500 N / m* and a mass of *m = 10 Kg*.

- What's the corresponding return force for a deformation of
*x = 2 m*? - What's the acceleration applied to the mass on that position?
- What's the corresponding period of oscillation?

Substituting into Hooke's law, we easily calculate the return force:

The corresponding momentarily acceleration is equal to:

Using the equation covered in the previous section, the period is:

which is independent of the amplitude and deformation. What changes is thus the maximum acceleration and velocity!

## RESOURCES:

### References

- https://www.britannica.com/science/simple-harmonic-motion
- https://courses.lumenlearning.com/boundless-physics/chapter/periodic-motion/
- https://openstax.org/books/physics/pages/5-5-simple-harmonic-motion

### Images

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

## 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

## Final words | Next up

And this is actually it for today's post!

Next time we will cover the energy in simple harmonic motion...

See ya!

Keep on drifting!

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