13.5: Summary

Key Takeaways

The equation of motion for the position, $x(t)$, of the mass in a one-dimensional spring-mass system with no friction can be written:

$$\begin{aligned} \frac{d^2x}{dt^2}=-\sqrt{\frac{k}{m}}x = -\omega^2 x\end{aligned}$$

and has a solution:

$$\begin{aligned} x(t) = A\cos(\omega t + \phi)\end{aligned}$$

where $A$ is the amplitude of the motion, $\phi$ is the phase, which depends on our choice of initial conditions (when we choose time $t=0$), and $\omega$:

$$\begin{aligned} \omega = \sqrt{\frac{k}{m}}\end{aligned}$$

is the angular frequency of the motion. The mass will oscillate about an equilibrium position with a period, $T$, and frequency, $f$, given by:

$$\begin{aligned} T&=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}}\\[4pt] f&=\frac{1}{T}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\end{aligned}$$

The velocity and acceleration of the mass are found by taking the time derivatives of the position $x(t)$:

$$\begin{aligned} x(t)&= A \cos(\omega t + \phi)\\[4pt] v(t)&=\frac{d}{dt}x(t) = -A\omega\sin(\omega t + \phi)\\[4pt] a(t)&= \frac{d^2}{dt^2}x(t) =\frac{d}{dt}\left( -A\omega\sin(\omega t + \phi)\right)= -A\omega^2\cos(\omega t + \phi)\end{aligned}$$

The total mechanical energy of the mass, at some position $x$, is given by:

$$\begin{aligned} E =U+K=\frac{1}{2}kx^2+\frac{1}{2}mv^2= \frac{1}{2}kA^2\end{aligned}$$

and is conserved.

Any system that can be described by the equation of motion:

$$\begin{aligned} \frac{d^2x}{dt^2}= -\omega^2 x\end{aligned}$$

is said to be a simple harmonic oscillator, and its position will be described by:

$$\begin{aligned} x(t) = A\cos(\omega t + \phi)\end{aligned}$$

A simple harmonic oscillator will always oscillate about an equilibrium position, where the net force on the oscillator is zero. The net force on a simple harmonic oscillator is always directed towards the equilibrium position, and has a magnitude proportional to the distance of the oscillator from its equilibrium position. The force is called a restoring force. A vertical spring-mass system, and a mass attached to two springs will both undergo simple harmonic motion about their respective equilibrium position.

A simple pendulum will undergo simple harmonic oscillations, if the amplitude of the oscillations is small. The angular frequency for the oscillations of a simple pendulum only depends on the length of the pendulum:

$$\begin{aligned} \omega = \sqrt{\frac{g}{L}}\end{aligned}$$

This is valid in the small angle approximation, where:

$$\begin{aligned} \sin\theta \approx \theta\end{aligned}$$

A physical pendulum of mass $m$ which oscillates about an axis through the object will also undergo simple harmonic oscillation in the small angle approximation. The angular frequency of the oscillations for a physical pendulum is given by:

$$\begin{aligned} \omega = \sqrt{\frac{mgh}{I}}\end{aligned}$$

where $h$ is the distance between the center of mass and the axis of rotation, and $I$ is the moment of inertia of the object about the rotation axis.

Important Equations

Position, velocity, and acceleration for SHM:

$$\begin{aligned} x(t)&= A \cos(\omega t + \phi)\\[4pt] v(t)&=\frac{d}{dt}x(t) = -A\omega\sin(\omega t + \phi)\\[4pt] a(t)&= \frac{d^2}{dt^2}x(t) = -A\omega^2\cos(\omega t + \phi)\end{aligned}$$

Period and frequency:

$$\begin{aligned} \omega &= \sqrt{\frac{k}{m}}\\[4pt] T&=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}}\\[4pt] f&=\frac{1}{T}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\end{aligned}$$

Mechanical energy:

$$\begin{aligned} E =U+K=\frac{1}{2}kx^2+\frac{1}{2}mv^2= \frac{1}{2}kA^2\end{aligned}$$

Simple pendulum (small angles):

$$\begin{aligned} \omega = \sqrt{\frac{g}{L}}\end{aligned}$$

Physical pendulum (small angles):

$$\begin{aligned} \omega = \sqrt{\frac{mgh}{I}}\end{aligned}$$

Important Definitions

Definition

Angular frequency: is related to a usual frequency by a factor of $2\pi$. For an object rotating around a circle at constant speed, the angular frequency of the rotation is the same as the angular speed (the rate of change of a position angle). SI units: $[\text{rad/s}]$. Common variable(s): $\omega$.