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Simple Harmonic Motion, Springs, and Pendulums

Mary Brown

Mary Brown

8 min read

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Study Guide Overview

This study guide covers Simple Harmonic Motion (SHM), including its definition, key characteristics (restoring force, displacement, angular frequency, period, velocity, acceleration, and energy), and relation to periodic motion. It explores SHM equations, graphs, and the generic differential equation. Examples of SHM in springs and pendulums are provided, along with formulas for their periods. The guide also discusses energy conservation in SHM and offers practice questions and exam tips.

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 Concept

Simple Harmonic Motion (SHM) is a special type of periodic motion where an object oscillates back and forth around a stable equilibrium position. Think of it like a perfectly swinging pendulum or a mass bouncing on a spring. The key is that the restoring force is always proportional to the displacement from equilibrium. 💡

Key Characteristics of SHM

  • Restoring Force: Always directed toward the equilibrium position and proportional to displacement (F=kxF = -kx).
  • Displacement: Described by a sine or cosine function: x(t)=Acos(ωt+ϕ)x(t) = A\cos(\omega t + \phi), where:
    • AA = Amplitude (maximum displacement)
    • ω\omega = Angular frequency
    • ϕ\phi = Phase angle (initial position)
  • Angular Frequency and Period: ω=2πT\omega = \frac{2\pi}{T}, where TT is the period. Frequency f=1Tf = \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)=Acos(ωt+ϕ)x(t) = A\cos(\omega t + \phi)

Velocity