mechanics-ai999x

About this model

mechanics-ai999x answers classical mechanics questions — damped, driven and coupled oscillations, physical and double pendulums, orbits and escape speed, moments of inertia, rolling, Atwood machines, precession, rotating frames, elastic collisions and rockets — with step-by-step working, units and LaTeX. It runs offline on any CPU as a Python package, with ONNX formula graphs for other languages. Version 2 adds search across every AI999X model — ask in your own words and it points you to the right formula, even in another model — and figures of any answer, drawn in your notebook.

Install

pip install mechanics-ai999x
hf download AI999X/mechanics-ai999x --local-dir mechanics-ai999x

Use

import mechanics_ai999x as mechanics
mechanics.ask("What is the escape speed from a planet of mass 5.972e24 kg and radius 6.371e6 m?")
# -> 11.2 km/s, with the formula and the steps

Formulas

25 capabilities. Open any of them on the capability orb to solve it in your browser.

Show all 25 formulas

Escape speed

v = \sqrt{\frac{2 G M}{r}}

Final velocity of the first body in a one-dimensional elastic collision

v_{1f} = \frac{\left(m_{1} - m_{2}\right) v_{1} + 2 m_{2} v_{2}}{m_{1} + m_{2}}

Final velocity of the second body in a one-dimensional elastic collision

v_{2f} = \frac{\left(m_{2} - m_{1}\right) v_{2} + 2 m_{1} v_{1}}{m_{1} + m_{2}}

Higher normal mode of a double pendulum

\omega = \sqrt{\frac{\left(2 + \sqrt{2}\right) g}{L}}

Higher normal-mode frequency of two coupled oscillators

\omega_{2} = \sqrt{\frac{k + 2 k_{c}}{m}}

Lower normal mode of a double pendulum

\omega = \sqrt{\frac{\left(2 - \sqrt{2}\right) g}{L}}

Moment of inertia of a solid disk about its axis

I_{r} = \frac{m R^{2}}{2}

Moment of inertia of a solid sphere

I_{r} = \frac{2 m R^{2}}{5}

Moment of inertia of a thin rod about its center

I_{r} = \frac{m L_{r}^{2}}{12}

Moment of inertia of a thin rod about one end

I_{r} = \frac{m L_{r}^{2}}{3}

Moment of inertia of a thin spherical shell

I_{r} = \frac{2 m R^{2}}{3}

Off-axis equilibrium of a bead on a rotating hoop

\theta = \arccos\left(\frac{g}{\omega^{2} R}\right)

Orbital period from the semi-major axis (Kepler's third law)

T = 2 \pi \sqrt{\frac{a^{3}}{G M}}

Orbital speed on an elliptical orbit (vis-viva)

v = \sqrt{G M \left(\frac{2}{r} - \frac{1}{a}\right)}

Period of a physical pendulum

T = 2 \pi \sqrt{\frac{I_{r}}{m g d}}

Precession period of a Foucault pendulum

T = \frac{2 \pi}{\Omega \sin\left(\phi\right)}

Quality factor of a damped oscillator

Q_{f} = \frac{\sqrt{k m}}{b}

Steady-state amplitude of a driven damped oscillator

A = \frac{F_{0}}{\sqrt{\left(k - m \omega^{2}\right)^{2} + \left(b \omega\right)^{2}}}

Velocity change of a rocket (Tsiolkovsky)

\Delta v = v_{e} \ln\left(\frac{m_{0}}{m_{f}}\right)

Topics

• Classical Mechanics
• Physics
• Step By Step Solutions
• Homework Help
• Education
• Offline
• Feature Extraction
• Semantic Search
• Physics Solver
• Mechanics

Versions

Files

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