Figure: Hooke's law and simple harmonic motion explicated

Image 1 Caption: A diagram illustrating ideal springs (i.e., those which obey the force law Hooke's law exactly) stretched to varying length. Hooke's law is the force law obeyed by real springs to one degree or another. However, Hooke's law (AKA the linear force law) is univerally important since it is the force law obeyed by almost all restoring forces which are the forces that establish stable equilibria of almost all kinds. Restoring forces cause simple harmonic motion in the ideal limit of NO dissipation of mechanical energy to waste heat or some other loss mechanism.

Further explication:

1. A harmonic oscillator is a system in which there is a restoring force that is linearly proportional to the displacement from mechanical equilibrium.

   The force law is called Hooke's law. The formula is

     F = - kx  , 

   where k is a constant which depends on the system, x is the displacement from mechanical equilibrium at x = 0, and the minus sign makes the force point opposite to the displacement.

   The minus sign makes the force a restoring force.

2. The potential energy (energy of position) of the Hooke's law force is given by

     V = (1/2)kx**2  . 

3. An ideal spring is an exact harmonic oscillator. Real springs approximate exact harmonic oscillators.

   

4. Image 2 Caption: A diagram illustrating simple harmonic motion (which is also sinusoidal motion) by an ideal spring which oscillated vertically. The horizontal axis is time.

   In simple harmonic motion , the absolute value of position is a maximum when velocity is zero and vice versa.

5. If a harmonic oscillator is subject only to the Hooke's law force, it undergoes simple harmonic motion which is illustrated in Image 2.

6. The differential equation for simple harmonic motion follows from Newton's 2nd law of motion (AKA F=ma) with Hooke's law providing the force:

     ma = m(d2x/dt2) = -kx  

   (where d²x/dt² is acceleration a) which has general solution

     x=A*sin(ωt)+B*cos(ωt)  , 

   where A and B are constants determined by the initial conditions and the angular frequency ω = sqrt(k/m). The solution consists of sinusoids (i.e., the sine function and the cosine function) as anticipated from the above description of simple harmonic motion.

7. Hooke's law is super-important because almost all sufficiently small displacements from stable equilibrium are subject to a Hooke's law restoring force. So Hooke's law is everywhere.

8. Hooke's law was discovered by Robert Hooke (1635--1703) (who is the eponym) in 1660, but NOT effectively published until 1678.