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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 talk about **Energy in Simple Harmonic Motion**.

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

## Forms of Energy in SHM

Let's continue from the horizontal mass-spring system model for simple harmonic motion that we described last time.

We already mentioned an equation which gives us the period (or frequency) from the mass *m* and *k* constant of the spring:

But, that's not quite enough! Being a motion, we of course want to know the velocity of the mass at each point of its oscillation, as well as its position as a function of time.

Let's try approaching this problem using conservation of energy...

### Potential Energy

The first form of energy is the elastic potential energy stored in the spring when its deformed. This energy is known and given by:

Its important to note that such energy is considered to be a form of conservative energy!

### Kinetic Energy

The second form of energy is the kinetic energy of the mass due to its motion, given by:

which is also conservative energy.

### Total Energy

Supposing that the return force of the spring is the one and only horizontal force applied on the system, and that the mass of the spring is negligible, the total energy of the system is thus conserved and given by:

### Maximum Energy

The total energy of the system depends on the amplitude *A* of its motion. Of course, when *x = ±A* the potential energy is max, and the velocity is zero (kinetic energy is zero), which yields:

### Maximum Velocity

Solving the previous equation for the velocity, now allows us to calculate the velocity, and taking it a step further the maximum velocity.

The velocity is given by:

For *x = 0* this equation gives us the maximum velocity:

## Simple Example

Let's consider a horizontal mass-spring system with a mass of *m = 0.5 Kg* and spring constant of *k = 300 N/m*. This system follows a SHM with amplitude *A = 0.05 m*. Calculate the following:

- Maximum energy of the system
- Maximum velocity
- Velocity at
*x = A / 2* - Maximum acceleration

The *k* constant and amplitude *A* are known quantities, and thus the first calculation is a simple substitution:

Similarly, the maximum velocity is also a simple substitution:

For *x = A / 2* we have to use the other equation, which gives us:

For the maximum acceleration we will use the equation from the previous post and substitute *x* with *A*, because the maximum return force and in turn acceleration is at the point of most deformation:

## RESOURCES:

### References

- https://courses.lumenlearning.com/physics/chapter/16-5-energy-and-the-simple-harmonic-oscillator/
- https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_I_-
*Mechanics_Sound_Oscillations_and_Waves*(OpenStax)/15%3A_Oscillations/15.03%3A_Energy_in_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

### Periodic Motion

- Periodic Motion Fundamentals -> Fundamentals (Period, Frequency, Angular Frequency, Return Force, Acceleration, Velocity, Amplitude), Simple Harmonic Motion, Example

## Final words | Next up

And this is actually it for today's post!

Next time we will get into the mathematical equations that describe simple harmonic motion...

See ya!

Keep on drifting!

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