Physics

Simple Harmonic Motion (S.H.M.)

Study of harmonic oscillators, position, velocity and acceleration equations, springs and pendulums.

  • Displacement (x or y): is the position of the particle with respect to the equilibrium position (with respect to the center of the trajectory). It is measured in m in the SI, and is represented by the letter x if the motion is horizontal, or by the letter y if it is vertical. Displacement or position is a function of time x(t) or y(t).
  • Amplitude (A): is the maximum displacement, or maximum separation from the equilibrium position, or also the distance from the end of the trajectory to its center. It is represented by the letter A. It is measured in (m).
  • Period (T): is the time it takes the particle to complete one full oscillation (s)
  • Frequency (f): is the number of full oscillations per unit of time. Or the inverse of the period. (Hz) or (s-1).
  • Angular frequency (ω\omega): is the rate at which the state of motion of the particle changes. (rad/s)
Characteristic parameters of harmonic oscillations
f=1Tω=2πf=2πTf = \frac{1}{T} \quad \omega = 2\pi f = \frac{2\pi}{T}
  • Phase (φ\varphi): The phase of the motion at any instant. It corresponds to the value φ=ω⋅t+φ0\varphi = \omega \cdot t + \varphi_0. It is the angle that represents the state of vibration of the body at a given instant. It has angle units, in the International System it is the radian (rad). When a full oscillation occurs, the phase increases by 2π2\pi radians and the body returns to its initial position (displacement) xx, since cos⁡φ=cos⁡(φ+2π)\cos \varphi = \cos(\varphi + 2\pi)
  • Initial phase (φ0\varphi_0): It is the angle that represents the initial state of vibration, that is, the displacement xx of the body at instant t=0t= 0. Its unit of measurement in the International System is the radian (rad)
The position of a particle with simple harmonic motion (s.h.m.), also called displacement, is determined by the distance x to the equilibrium position. It is a sinusoidal function (sine or cosine), which depends on time x=f(t)x = f(t).
x(t)=A⋅sin⁡(ω⋅t+φ0)orx(t)=A⋅cos⁡(ω⋅t+φ0)x(t) = A \cdot \sin(\omega \cdot t + \varphi_0) \quad \text{or} \quad x(t) = A \cdot \cos(\omega \cdot t + \varphi_0)
Assuming an initial phase equal to 0, φ0=0\varphi_0=0, and using the sine form, the graph of position with respect to time is the following:
Position vs time graph
The instantaneous velocity of the particle at each instant is given by the derivative of the position with respect to time. Using the sine expression for x(t):
v(t)=dxdt=A⋅ω⋅cos⁡(ω⋅t+φ0)v(t) = \frac{dx}{dt} = A \cdot \omega \cdot \cos(\omega \cdot t + \varphi_0)
The maximum velocity is A⋅ωA \cdot \omega, with cos⁡(ω⋅t+φ0)=±1\cos(\omega \cdot t + \varphi_0) = \pm 1 when the particle passes through the equilibrium position.

The instantaneous acceleration of the particle at each instant is given by the derivative of the velocity:
a(t)=dvdt=−A⋅ω2⋅sin⁡(ω⋅t+φ0)a(t) = \frac{dv}{dt} = -A \cdot \omega^2 \cdot \sin(\omega \cdot t + \varphi_0)
The maximum acceleration is A⋅ω2A \cdot \omega^2, with sin⁡(ω⋅t+φ0)=±1\sin(\omega \cdot t + \varphi_0) = \pm 1, when the particle passes through the extremes.

Relationships between position, velocity and acceleration:
v=±ωA2−x2a=−ω2xv = \pm \omega \sqrt{A^2 - x^2} \quad \quad a = -\omega^2 x
Mass-spring system
The elastic spring is a harmonic oscillator. The force it exerts on the attached mass is proportional to the displacement:
F=−k⋅xF = -k \cdot x
The angular frequency and period of oscillation are given by:
ω2=kmT=2⋅πmk\omega^2 = \frac{k}{m} \quad \quad T = 2\cdot \pi \sqrt{\frac{m}{k}}
The elastic potential energy accumulated by a spring is:
Us=12k⋅x2U_s = \frac{1}{2} k \cdot x^2
The mechanical energy of the system is:
E=Us+K=12k⋅x2+12k⋅(A2−x2)=12k⋅A2E = U_s + K = \frac{1}{2} k \cdot x^2 + \frac{1}{2} k \cdot (A^2 - x^2) = \frac{1}{2} k \cdot A^2
Simple pendulum
A simple pendulum behaves like a harmonic oscillator when it oscillates with small amplitudes. The acceleration is proportional to the displacement but in the opposite direction, with the expression:
a=−glxa = -\frac{g}{l} x
The period for small oscillations depends only on the length. Neither the mass of the oscillating body nor the amplitude of the oscillation influences it.
T=2πlgT = 2\pi \sqrt{\frac{l}{g}}