Oscillation

Angular Motion

Trigonometry

• sine, cosine, tangent

Polar vs. Cartesian Coordinates

• Cartesian Coordinates are named for René Descartes, the French mathematician who developed the ideas. It describes a position using two orthogonal base axes.
• Polar Coordinates describe a position using an angle and a radius.

Oscillation Amplitude and Period

• Simple harmonic motion (or "the periodic sinusoidal oscillation of an object") describes motion that follows a sine wave.
  • Amplitude: The distance from the center of motion to either extreme
  • Period: The amount of time it takes for one complete cycle of motion
  • Frequency: 1/period
• Waves with different amplitude and frequency can be mixed together, just like octaves for noise.

Pendulum

1. If we look at the pendulum itself, the only force it experiences along the tangent direction is the component of gravity pointing along the tangent, which is $F_p = F_g * sin(\theta)$.
2. Given that when $\alpha$ is super small, $sin(\alpha) \approx \alpha$ (where $\alpha$ is in radians), and $sin(\alpha) \approx \frac{circleSegment}{radius}$. We know that $angularSpeed = \frac{linearSpeed}{radius}$, and similarly $angularAcceleration = \frac{linearAcceleration}{radius}$.
3. So $angularAcceleration = \frac{|\vec{a}|}{radius} = \frac{|\vec{F_p}| / m}{radius} = \frac{|\vec{F_g}| * sin(\theta) / m}{radius}$

Springs

Hooke's law

The force of the spring is directly proportional to the extension of the spring. $F_{spring} = -k * x$, where: $k$ is a constant property of the spring $x$ is the displacement of the spring