Masses connected to ideal springs and simple pendulums are examples of Simple Harmonic Oscillators. When set into motion the masses will oscillate back and forth in a predictable sinusoidal way. These two systems provide excellent case studies in SHO.
Check out the mesmerizing Double Pendulum
Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.
A Simple Harmonic Oscillator is a system where the restoring force acting on the object is linear with respect to displacement and it's potential energy is quadratic with respect to displacement. Masses connected to springs and simple pendulums behave this way and thus provide excellent case studies into the features of SHO.
--Springs--
Recall the force acting on an object fixed to the end of a spring is $|F^s| = k |\Delta x|$, where x is the displacement from equilibrium and k is the spring constant. Also recall that the potential energy is $U^s=\frac{1}{2}kx^2$. Here the displacement variable is a position measured by distance. The kinematic equations describing the motion are sinusodial, as expected for all SHO. In general, all SHO have an angular velocity $\omega=\frac{2 \pi}{T}$, where T is the period. For a spring specific system, there is also the useful relationship below.
$\omega=\sqrt{\frac{k}{m}}$, where k = spring constant and m = mass
Here you see $\omega$ (and thus T) is independent of the energy of the system. This is an expected feature among SHO where the period is independent of the amplitude, and thus energy, of the oscillation.
--Simple Pendulums--
A mass connected to a light string - a simple pendulum - that oscillates at small angles ($\theta < 10^0$), the net force of gravity and the string act as approximately a linear restoring force. This means the pendulum behaves like a SHO. The natural variable to track the position of the pendulum is the angle $\theta$, measured with respect to the vertical. With this as the displacement variable, the motion follows a sinusoidal form. An example for a pendulum released from rest from a positive maximum position ($\theta_{max}$) at t = 0, this would look like the following,
$\theta (t) = \theta_{max} cos(\omega t)$, where $\omega$ is the angular velocity.
All SHO have the relationship that $\omega = \frac{2 \pi}{T}$, where T is the period, but pendulums also have a useful relationship between $\omega$ and the length of the pendulum (l), and the gravitational field (g) they pendulum resides in.
$\omega=\sqrt{\frac{g}{l}}$
Here you see $\omega$ (and thus T) is still independent of the energy of the system but also independent of the mass connected to the object. If the pendulum is raised to angles greater than 10 or 15 degrees, it no longer behaves like a SHO and the motion requires additional considerations.
Now, take a look at the pre-lecture reading and videos below.
