Physics in Motion

# Dynamics of Simple Harmonic Motion

Conservative forces can be described through a potential function. In one-dimension, the relation between the force and kinetic energy is given by,

$$F = - \frac{d U(x)}{dx}$$

The simplest function is $$U(x) = constant$$, which results in no force.

The next simplest function is a linear function, $$U(x) = F_c x$$, with $$F_c$$ some constant with dimensions of Newton. This potential results in a constant force $$F = -F_c$$, and a constant acceleration $$a = - F_c / m$$. We have encountered such a potential in the case of gravity near the surface of the earth where $$F_c = m g$$, and $$x$$ was a measure of the height from the surface, $$h$$. In that case the potential energy is simply $$U(h) = mgh$$.

At the next order of complexity is a quadratic function, $$U(x) = \frac{1}{2} k x^2$$, where $$k$$ is some constant with units of $$Newton/m$$. This is a very important potential that appears in a wide variety of phenomena. The force associated with such a potential is,

$$F = - \frac{d U(x)}{d x} = - k x(t)$$

You might recognize this force as that of a simple spring with spring constant $$k$$, namely Hooke's law. This is indeed the case, but it turns out that this force also serves as a good model for many other physical systems. The equation of motion associated with such a force is,

$$\frac{d^2 x(t)}{d t^2} = - \frac{k}{m} x(t)$$

As usual, this is a differential equation whose solution is the position as a function of time, $$x(t)$$. The general solution to the equation of motion is given by,

$$x(t) = A \cos (\omega t + \phi)$$

where

$$\omega = \sqrt{k/m}$$

is called the angular frequency. It is an intrinsic quantity of the system. $$A$$ is the amplitude of the oscillations of $$\phi$$ is the phase, which defines the initial position. Both $$A$$ and $$\phi$$ are extrinsic quantities, which vary depending on the initial conditions.

Such a motion is known as simple harmonic motion. It is an extremely important system, not just because of its simplicity, but because it shows up in a countless number of situations in all fields of science.

The motion is periodic with a period of $$T = 2 \pi / \omega$$. To see this we notes that,

$$x(t + T) = A \cos (\omega (t + T) + \phi)$$

$$x(t + T) = A ( \cos(\omega T) \cos(\omega t + \phi) - \sin(\omega T) \sin(\omega T + \phi) )$$

$$x(t + T) = A \cos(\omega t + \phi)$$

$$x(t + T) = x(t)$$

The frequency is the inverse of the period $$f = 1 / T$$ and hence we have a relation between the frequency and the angular frequency,

$$\omega = 2 \pi f = \frac{2 \pi}{T}$$

Since we have the full solution for the position as a function of time, we can obtain all the other kinematical functions such as velocity and acceleration through differentiation,

$$v(t) = \frac{d x(t)}{d t} = - A \omega \sin(\omega t + \phi)$$

$$a(t) = \frac{d v(t)}{d t} = - A \omega^2 \cos(\omega t + \phi)$$

Since the force is derived from a potential, it is a conservative force and thus the total energy is a conserved quantity. As usually, it is composed of kinetic energy and a potential energy,

$$K(t) = \frac{1}{2} m v(t)^2 = \frac{1}{2} m A^2 \omega^2 \sin^2 (\omega t + \phi)$$

$$U(t) = \frac{1}{2} k x(t)^2 = \frac{1}{2} k A^2 \cos^2 (\omega t + \phi)$$

Each of these quantities varies with time, but their sum, which is the total energy, is a constant of the motion:

$$E_{tot} = K(t) + U(t) = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2$$

This last equation is proved using the relation $$\omega = \sqrt{k / m}$$ and the basic identity of trigonometry.