## 1. Figure out what type of system we are working with

First things first, we need to know what type of system we are working with.

There are usually 3 types:

- Mass on a spring
- Simple Pendulum
- Physical Pendulum

Once we know what system we are working with, we can pull the equations that we need off of our formula sheet.

## 2. Determine the Angular Frequency and the Equilibrium Position

Let’s start with the easy one, the equilibrium position.

The equilibrium position is typically given in the problem and it is usually zero.

For a mass on a spring, the equilibrium position is going to be where the spring is unstretched.

For a pendulum, the equilibrium position is going to be where the pendulum is oriented vertically.

Now, for the angular frequency.

Usually, we can calculate the frequency from the system-specific equations on our formula sheet.

Sometimes they will give us information about the frequency or period and we can use that to calculate the angular frequency.

## 3. Calculate the Amplitude and Phase Shift using the initial conditions

There are usually three types of initial conditions for Simple Harmonic Motion problems:

- Non-zero initial position and zero initial velocity
- Zero initial position and non-zero initial velocity
- Non-zero initial potion and non-zero initial velocity

In any case, use the initial conditions with the equations for position and velocity at a time equal to zero.

This will give you two equations with two unknowns that you can use to solve for the amplitude and phase shift.

The first two cases will be fairly straightforward but the third case will be a little harder.

Check out this example problem that shows case 2.

Check out this example problem that shows case 3.

## 4. Determine our equations for position, velocity, and acceleration

Now the hardest part is over.

Use the amplitude, angular frequency, phase shift, and equilibrium position to come up with an equation for the position.

Next, take the derivative with respect to time to determine an equation for velocity.

Lastly, take another derivative with respect to time to determine an equation for acceleration.

## 5. Calculate the kinematics or energy of the system

Now that we have our equations set up, we can solve what they are asking for in the problem.

They may ask to solve for the position, velocity, or acceleration, at a certain time.

Or they could ask for the energy of the system at a certain time.

Check out the YouTube videos below to see examples of each.

## Check out these YouTube videos to see solved examples

When I was in engineering school, I always started studying by watching YouTube videos (shoutout The Organic Chemistry Tutor).

Seeing other people solve problems helped me understand the thought process behind the steps to the solution.

Once I got the basics, I would start solving practice problems myself.

I am working on a Physics video series on YouTube where I go through theory and practice problems.

Check out these videos below!

## Simple Harmonic Motion Videos

## Want extra practice problems with solutions? Check out my eBooks

Once you have watched a couple of YouTube videos, it’s time to practice for yourself!

Check out these physics eBooks with practice problems and full solutions.

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