KAHIBARO
Table of Contents
The Mathematical Form of Simple Harmonic Motion
Simple harmonic motion is a special kind of periodic motion in which the restoring effect always points toward equilibrium and is proportional to displacement. The SHM equation is the mathematical expression that describes how the displacement changes with time.
If the displacement from equilibrium is called $x(t)$, then the most common form of the SHM equation is
$$
x(t) = A \cos(\omega t + \phi)
$$
A completely equivalent form is
$$
x(t) = A \sin(\omega t + \phi)
$$
Both describe the same kind of motion. The difference is only in the choice of starting point in time.
Meaning of the Symbols
Each symbol in the equation has a clear physical meaning.
| Symbol | Meaning | Unit |
|---|---|---|
| $x(t)$ | displacement at time $t$ | m |
| $A$ | amplitude, the maximum displacement | m |
| $\omega$ | angular frequency | rad/s |
| $t$ | time | s |
| $\phi$ | phase constant, also called initial phase | rad |
The amplitude $A$ tells us how far the object moves from equilibrium. The angular frequency $\omega$ tells us how fast the oscillation happens. The phase constant $\phi$ tells us where in the cycle the motion starts at $t = 0$.
For SHM, the displacement must vary sinusoidally with time:
$$
x(t) = A \cos(\omega t + \phi)
$$
or
$$
x(t) = A \sin(\omega t + \phi)
$$
This is the defining mathematical form of simple harmonic motion.
Why Sine and Cosine Appear
Sine and cosine functions repeat regularly, so they are natural choices for periodic motion. In SHM, the object moves back and forth in a smooth and symmetric way. The displacement is largest at the ends, zero at equilibrium, and repeats after each cycle. Sinusoidal functions match this pattern perfectly.
The choice between sine and cosine depends on the initial condition. For example, if the object starts at maximum displacement at $t = 0$, cosine is often convenient because $\cos 0 = 1$.
Connection with Period and Frequency
Angular frequency is related to the ordinary frequency $f$ and the period $T$ by
$$
\omega = 2\pi f = \frac{2\pi}{T}
$$
So the SHM equation can also be written as
$$
x(t) = A \cos\left(2\pi f t + \phi\right)
$$
or
$$
x(t) = A \cos\left(\frac{2\pi}{T} t + \phi\right)
$$
These forms are useful when the period or frequency is known directly.
Important relationships for SHM:
$$
\omega = 2\pi f
$$
$$
\omega = \frac{2\pi}{T}
$$
$$
T = \frac{2\pi}{\omega}
$$
$$
f = \frac{\omega}{2\pi}
$$
Initial Conditions and Phase Constant
The phase constant $\phi$ determines the starting position in the cycle. At time $t = 0$,
$$
x(0) = A \cos \phi
$$
This means that the initial displacement depends on $\phi$.
A few common cases are especially useful.
| Phase constant | Initial displacement | Interpretation |
|---|---|---|
| $\phi = 0$ | $x(0) = A$ | starts at positive maximum |
| $\phi = \pi$ | $x(0) = -A$ | starts at negative maximum |
| $\phi = \frac{\pi}{2}$ | $x(0) = 0$ for cosine form | starts at equilibrium |
| $\phi = -\frac{\pi}{2}$ | $x(0) = 0$ for cosine form | starts at equilibrium in opposite direction |
So the phase constant is not an extra decoration. It is what allows the same equation to describe many different starting situations.
Velocity and Acceleration from the SHM Equation
The SHM equation gives displacement as a function of time. Velocity and acceleration follow by differentiation.
Starting from
$$
x(t) = A \cos(\omega t + \phi)
$$
the velocity is
$$
v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi)
$$
and the acceleration is
$$
a(t) = \frac{d^2x}{dt^2} = -A\omega^2 \cos(\omega t + \phi)
$$
Since $x(t) = A \cos(\omega t + \phi)$, this becomes
$$
a(t) = -\omega^2 x(t)
$$
This is one of the most important equations in SHM.
In simple harmonic motion, acceleration is proportional to displacement and opposite in direction:
$$
a = -\omega^2 x
$$
The minus sign shows that the acceleration always points toward the equilibrium position.
Maximum Values
The displacement reaches maximum magnitude at the turning points, where $x = \pm A$. The maximum speed and acceleration can be read from the equations.
From
$$
v(t) = -A\omega \sin(\omega t + \phi)
$$
the largest possible speed is
$$
v_{\max} = A\omega
$$
From
$$
a(t) = -A\omega^2 \cos(\omega t + \phi)
$$
the largest possible acceleration magnitude is
$$
a_{\max} = A\omega^2
$$
These values are useful because they connect the amplitude and angular frequency to the most extreme motion.
How the Motion Changes During One Cycle
The SHM equation describes a repeating pattern. If we use
$$
x(t) = A \cos(\omega t)
$$
for simplicity, then one full cycle looks like this.
| Time | Displacement |
|---|---|
| $t = 0$ | $x = A$ |
| $t = T/4$ | $x = 0$ |
| $t = T/2$ | $x = -A$ |
| $t = 3T/4$ | $x = 0$ |
| $t = T$ | $x = A$ |
This shows the repeating nature of the sinusoidal function.
Graphical Picture
The displacement versus time graph for SHM is a smooth cosine or sine curve.
The graph repeats every period $T$. The highest points are $+A$, the lowest points are $-A$, and the curve crosses zero at the equilibrium position.
Equivalent Forms of the SHM Equation
The SHM equation can be written in several equivalent ways:
$$
x(t) = A \cos(\omega t + \phi)
$$
$$
x(t) = A \sin(\omega t + \phi)
$$
$$
x(t) = C \cos(\omega t) + D \sin(\omega t)
$$
These are mathematically interchangeable. The last form is especially useful when combining information about initial displacement and initial velocity.
The Differential Equation Behind SHM
The sinusoidal form is not arbitrary. It comes from the defining differential equation of simple harmonic motion:
$$
\frac{d^2x}{dt^2} + \omega^2 x = 0
$$
Any function that satisfies this equation represents SHM. Sine and cosine satisfy it exactly, which is why they appear in the motion equation.
The standard differential equation for simple harmonic motion is
$$
\frac{d^2x}{dt^2} + \omega^2 x = 0
$$
Its solutions are sinusoidal functions of time.
A Simple Example
Suppose an object oscillates with amplitude $0.20 \, \text{m}$ and angular frequency $4 \, \text{rad/s}$, starting at maximum displacement. Then $\phi = 0$, so
$$
x(t) = 0.20 \cos(4t)
$$
Its velocity is
$$
v(t) = -0.80 \sin(4t)
$$
and its acceleration is
$$
a(t) = -3.2 \cos(4t)
$$
The period is
$$
T = \frac{2\pi}{4} = \frac{\pi}{2} \, \text{s}
$$
So the entire motion repeats every $\frac{\pi}{2}$ seconds.
What the Equation Tells Us Physically
The SHM equation contains the whole time behavior of the oscillation. It tells us where the object is at any moment, how large the motion is, how fast the repeating occurs, and how the motion begins. Once $A$, $\omega$, and $\phi$ are known, the motion is completely determined.
This equation is one of the most important mathematical models in physics because many systems, at least approximately, oscillate in this way near equilibrium.