These are my complete notes for Oscillations in Classical Mechanics.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Oscillations (Classical Mechanics)
XI. Oscillations
XI.I Simple Harmonic Motion.
# P. Rule 152. A Simple Harmonic Motion system is one that will continue to oscillate back and forth forever in the absense of friction. The main two types of SHM are pendulums and springs.
Any force can cause simple harmonic motion, as long as it passes these two requirements:
- The force causing simple harmonic motion must be a restoring force.
- The magnitude of the restoring force must be proportional to the displacement from equilibrium position. For example, the Law of Elasticity (and thus all springs) follows this specification.
# Simple Harmonic Motion from the perspective of the three base positions:
The initial position of the pendulum or spring weight being used for SHM, prior to being released, will be position 1. When released, the weight will move through position 2 (the rest position) and slow into position 3 (the maximum displacement on the other side of position 2). Afterwards, the weight will returns backward through position 2 into position 1, restarting the cycle.
The velocities at position one and three are equal to zero. The velocity is at its maximum at position two.
The acceleration will be linearly proportional to the spring force, meaning it follows the same pattern as the spring force. Thus, the magnitude of acceleration will be at it's max at positions 1 & 3, while the acceleration at position 2 will be 0.
When moving toward rest position, acceleration and velocity are in the same direction and thus the weight is speeding up. When moving away from rest position, however, the weight will now be slowing down, as the acceleration is always pointed toward the rest position (as it is tied to the force) and velocity away from it.
# P. Rule 153. Simple Harmonic Motion is not Uniformly Accelerated Motion. You can't use the UAM equations for SHM. Live with it, punk.
# Amplitude: The magnitude of the maximum displacement from the equilibrium position, during simple harmonic motion.
# P. Rule 154. The difference between horizontal and vertical mass-spring systems lies solely in how the rest position changes in a vertical system due to gravity pulling the weight downward. In practice, the equations and everything will remain the same, vertically or horizontally, as the vertical system will still oscillate around that new rest point created by the weight.
# P. Rule 155. Pendulum Oscillation:
A pendulum is only considered to be in Simple Harmonic Motion when has an amplitude less than 15 degrees, as a result of the restoring force of pendulums. The restoring force for a pendulum is the component of gravity in the tangential direction, hereafter referred to as "the force of gravity tangential". This has the equation mgsin(θ).
θ being the angle made between the swing of the pendulum with the rest position. As the angle increases (e.g., the displacement from equilibrium position), the force of the gravity tangential increases. The force of gravity tangential is at 0 when the pendulum is at resting position.
As the angular displacement from rest position increases, the force of gravity tangential will increasingly point to a location below the rest position. This is patently not the rest position. Therefore, the force of gravity tangential is only a restoring force for small angles - namely, for oscillations where the amplitude < 15° (due to the small angle approximation - [[[[).
# P. Rule 156.
In a mass-spring system, the displacement from the rest position is a linear displacement, ∆x.
In a pendulum system, the displacement from the rest position is an angular displacement, θ. This is the result of the pendulum oscillating through a small arc-length of a circle, given it is oscillating at it's maximum extension. Use radians for the units.
