#AP Physics C: Mechanics - Simple Harmonic Motion (SHM) Study Guide 🚀

Hey there, future physics master! Let's dive into Simple Harmonic Motion (SHM). This guide is designed to be your go-to resource, especially the night before the exam. We'll break down everything you need to know, from the basics to the trickiest applications. Let's get started!

#What is Simple Harmonic Motion?

#Key Characteristics of SHM

• Restoring Force: Always directed toward the equilibrium position and proportional to displacement ($F = -kx$).
• Displacement: Described by a sine or cosine function: $x(t) = A\cos(\omega t + \phi)$, where:
  • $A$ = Amplitude (maximum displacement)
  • $\omega$ = Angular frequency
  • $\phi$ = Phase angle (initial position)
• Angular Frequency and Period: $\omega = \frac{2\pi}{T}$, where $T$ is the period. Frequency $f = \frac{1}{T}$.
• Velocity and Acceleration: Also sinusoidal, with maximum values at the equilibrium position.
• Energy: A continuous exchange between kinetic and potential energy, with total mechanical energy conserved.
• Ubiquitous: Found in springs, pendulums, and even electrical circuits.
• Idealized: Real-world SHM may be affected by damping and external forces.

#Understanding Periodic Motion

Before we dive deeper into SHM, let's define periodic motion. It's any motion that repeats itself over a consistent time interval. SHM is a specific type of periodic motion, characterized by that special restoring force.

#SHM Equations: Position, Velocity, and Acceleration

Here's where calculus comes into play! Remember, velocity is the derivative of position, and acceleration is the derivative of velocity. This is how we get the following equations:

#Position

$$x(t) = A\cos(\omega t + \phi)$$

#Velocity