Αποτελέσματα Αναζήτησης
1. One of the most important examples of periodic motion is simple harmonic motion (SHM), in which some physical quantity varies sinusoidally. Suppose a function of time has the form of a sine wave function, y(t) = Asin(2πt / T ) (23.1.1) where A > 0 is the amplitude (maximum value).
An example of a simple harmonic oscillator is the simple pendulum, as shown in the diagram on the right. The pendulum oscillates around a central midpoint known as the equilibrium position. Marked on the diagram by an x is the measure of displacement, and by an A is the amplitude of the oscillations - this is the maximum displacement.
The characteristics of simple harmonic motion include: A force (and therefore an acceleration) that is opposite in direction, and proportional to, the displacement of the system from equilibrium. Such a force, that acts to restore the system to equilibrium, is known as a restoring force. No loss of mechanical energy.
In these notes, we introduce simple harmonic oscillator motions, its defining equation of motion, and the corresponding general solutions. We discuss how the equation of motion of the pendulum approximates the simple harmonic oscillator equation of motion in the small angle approximation.
We’ll start with γ = 0 and F = 0, in which case it’s a simple harmonic oscillator (Section 2). Then we’ll add γ, to get a damped harmonic oscillator (Section 4). Then add F(t) (Lecture 2). The damped, driven oscillator is governed by a linear differential equation (Section 5).
Simple harmonic motion is a type of oscillation, where the acceleration of the oscillator is directly proportional to the displacement from the equilibrium position, and acts towards the equilibrium position.
Figure 4.1: Harmonic Oscillation of a mass at a spring. At the maximum elongation the spring is pulling on the mass. The mass gets accelerated towards the equilibrium position. At the equilibrium position the acceleration is zero and the velocity of the mass reaches its maximum.
