SHM Max Speed⚠ unverified
Trigonometry / Applications · Maximum speed in simple harmonic motion
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| A | A | m | 2 | Amplitude |
| omega | ω | radian / s | 3.0 | Angular frequency |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| vmax | vmax | m/s | Maximum speed |
The science & history
Understanding the Parameters
- A (m) — Amplitude.
- ω (radian / s) — Angular frequency.
- Output vmax (m/s) — Maximum speed.
How to Calculate
- Enter Amplitude as
A(default 2 m). Use the unit menu when you need a different unit. - Enter Angular frequency as
omega(default 3.0 radian / s). Use the unit menu when you need a different unit. - Click Calculate. The card evaluates $v_{\max}=A\omega$ 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 m/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 Period