13.5: Summary
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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): \(\omega\) .