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.