warehouse/content/physics/harmonic-motion.md

title, date

title	date
Less Simple Harmonic Motion	2025-01-09

Simple harmonic motion - Spring oscillations

When an object vibrates or oscillates back and forth, over the same path, each oscillation taking the same amount of time, the motion is periodic. The simplest form of periodic motion closely resemble this system, we will examine ... blah blah blah. Any spring at natural length is defined to be at its equilibrium position. If additional energy is imparted upon the system, either compressing or stretching said spring, a restoring force will occur to restore the initial position.

F=\pm kx

Terms, I guess:

• the distance from the equilibrium point is displacement
• Peak displacement is called the amplitude
• one cycle is the distance covered between the trough of the position curve over time and the peak of the same curve
• the period T is the time required to complete one cycle's worth of motion
• the frequency f is the number of complete cycles per second.

f=\frac{1}{T} \text{ and } T=\frac{1}{f}

Energy in Simple Harmonic Motion

Please just go learn Lagrangian Mechanics already. The energy approach is obviously the most useful of them all. Elastic Potential energy:

|\int{F\mathrm{d}x}|=\int{kx \mathrm{d}x}=\tfrac{1}{2}kx^2 

The total energy: E=\textbf{KE}+\textbf{PE}

E=\tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 

We all know what x and v are. SHM can only occur if friction and other external forces are approximately net zero. A is defined as the absolute peak distance from equilibrium. When x=\pm A, the \textbf{KE}=0.

\tfrac{1}{2}m0^2+\tfrac{1}{2}kA^2 

At equilibrium, velocity and therefore kinetic energy are the greatest, with \textbf{PE}=0:

E=\tfrac{1}{2}m(v_{\text{max}})^2 

Since energy is always conserved in SHM:

\tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2 

Solving the system of equations for v:

v^2 = \frac{k}{m}A^2(1-\frac{x^2}{A^2})

Given mv^2=kA^2:

v_max=\sqrt{\frac{k}{m}}A 

With another step of substitution:

v=\pm v_{\text{max}}\sqrt{1-\frac{x^2}{A^2}}

The Period and the Sinusoidal Nature of SHM

The period of harmonic motion depends on the stiffness of the spring and the mass m in question. It does not, however, depend on the amplitude ,$A$.

T=2\pi\sqrt{\tfrac{m}{k}}

Larger mass means more rotational inertial (we haven't gotten to this yet), so that part of it makes sense. It is also important that k\propto F

f=\frac{1}{t}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}