SHM Period⚠ unverified
Trigonometry / Applications · Period of simple harmonic motion
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| omega | ω | radian / s | 3.1415926535 | Angular frequency |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| T | T | s | Period |
The science & history
Understanding the Parameters
- ω (radian / s) — Angular frequency.
- Output T (s) — Period.
How to Calculate
- Enter Angular frequency as
omega(default 3.1415926535 radian / s). Use the unit menu when you need a different unit. - Click Calculate. The card evaluates $T=2\pi/\omega$ and shows the result in the declared output unit; Result in (when present) converts that number.
The Science: A Rigorous Derivation (Approaching a Proof)
Angle of elevation converts a surveyed baseline into a height by $\tan\theta = h/d$. Simple harmonic motion $x = A\cos(\omega t+\phi)$ is the projection of uniform circular motion: period $2\pi/\omega$, peak speed $A\omega$. Both are the right-triangle (or unit-circle) definitions applied to a physical situation, not new trigonometry.
Dimensional check. The declared output unit is s; ToolBox evaluates the
formula in SI (radians internally for every trigonometric call) and python-calc converts to
the unit on the card.
History and Development
Height-and-distance problems are in every early-modern practical geometry (Gemma Frisius, the surveyor's theodolite). SHM is Huygens and Newton; the circular-motion projection picture is already in Galileo's notes on the pendulum.
Related Concepts: Height From Elevation, Distance From Elevation, SHM Max Speed