classical-physics-ai999x — offline, verified classical physics solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Physical Sciences · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

classical-physics-ai999x is an offline, deterministic Python solver that answers classical physics questions with verified results from 196 certified capabilities, gives 1177 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

classical-physics-ai999x is a compact, deterministic classical physics engine for Python. It reads a question in plain English (or a typed call), picks a certified classical physics capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified classical physics capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 1177 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: kinematics, dynamics, energy, momentum, rotation, gravitation, oscillations, waves and sound, optics, thermodynamics, electricity and magnetism.

Model overview

Property	Value
Package	classical-physics-ai999x 1.0.0 (import classical_physics_ai999x)
Type	Deterministic neuro-symbolic classical physics engine (rule-based reading + certified symbolic capabilities)
Capabilities	46 certified derivations, 16 certified model_familys, 2 certified procedures, 132 certified relations
Knowledge pack	version 1.0.2
Golden answers	1177 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install classical-physics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/classical-physics-ai999x/resolve/main/dist/classical_physics_ai999x-1.0.0-py3-none-any.whl

import classical_physics_ai999x as classical_physics

classical_physics.ask("A 3 kg object moves at 4 m/s. What is its kinetic energy?").show()
# -> SOLVED — 24 J  (kinetic_energy)
classical_physics.ask("A 2 kg mass on a spring with a spring constant of 50 N/m has a damping coefficient of 4 kg/s. What is the damped angular frequency?").show()
# -> SOLVED — 4.89898 rad/s  (damped_angular_frequency)
classical_physics.ask("A ball is kicked at 20 m/s at 30 degrees above the horizontal from level ground. How far does it travel horizontally?").show()
# -> SOLVED — 35.324 m  (projectile)
classical_physics.ask("What is the escape speed from a planet of mass 5.972e24 kg and radius 6.371e6 m?").show()
# -> SOLVED — 11186 m/s  (escape_speed)
classical_physics.solve("rocket_delta_v", ve="3000 m/s", m0="1000 kg", mf="400 kg").show()
# -> SOLVED — 2748.87 m/s  (rocket_delta_v)
classical_physics.ask("Who created you?").message
# -> classical-physics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: classical-physics-ai999x "<question>", classical-physics-ai999x --capabilities. The outputs above were produced by running this exact code on classical-physics-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

46 certified derivations, 16 certified model_familys, 2 certified procedures, 132 certified relations. A few of them:

• Derivations: acceleration with friction level; angular acceleration from tangential; angular frequency period; average speed equal instantaneous; centripetal acceleration from period; derive absolute pressure depth; derive acceleration frictionless incline; derive centripetal acceleration omega; derive centripetal force; derive charge current; derive cyclotron radius; derive electric power; derive impulse; derive inductive reactance; derive lc resonance; derive long wire field; derive normal on level; derive ohm power current; derive ohm power voltage; derive orbital period; derive photon energy; derive photon energy wavelength; derive photon momentum; derive rl time constant …
• Model_familys: A gas between two states; Apparent weight in an elevator; Block on a surface pulled over a pulley; Collisions along a line; Doppler effect for sound; Ideal banking of a curve; Maximum speed on a flat curve; Mixing to thermal equilibrium; Motion in stages; Projectile motion; RC circuits; Radioactive decay; Recoil from rest; Rolling from a height; Series and parallel resistor and capacitor networks; Two masses over a pulley
• Procedures: Two resistors in series and in parallel; Velocity and displacement after constant acceleration
• Relations: Absolute pressure at a depth in a fluid; Acceleration down a frictionless incline; Acceleration of a body rolling without slipping down an incline; Acceleration of an Atwood machine with a massive pulley; Amplitude-resonance frequency of a driven damped oscillator; Angular frequency and frequency; Angular frequency of a damped oscillator; Angular momentum of a point mass on a circle; Angular momentum of a rotating body; Angular velocity and angle turned under constant angular acceleration; Angular velocity under constant angular acceleration; Average net force from a change of momentum; Average power; Body-frame precession rate of a torque-free symmetric body; Capacitance of parallel plates; Carnot (maximum) efficiency; Centripetal acceleration; Centripetal acceleration from angular velocity; Centripetal force; Charge carried by a steady current; Charge on a capacitor; Coefficient of performance of a heat pump; Coefficient of performance of a refrigerator; Component of the weight along an incline …

classical_physics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 134 formulas

Show every formula, by topic

AC circuits

Reactance of a capacitor

$$X=\frac{1}{2\pi fC}X\; =\; \backslash frac\{1\}\{2\; \backslash pi\; f\; C\}$$

Reactance of an inductor

$$X=2\pi fLX\; =\; 2\; \backslash pi\; f\; L$$

Resonant frequency of an LC circuit

$$f=\frac{1}{2\pi \sqrt{LC}}f\; =\; \backslash frac\{1\}\{2\; \backslash pi\; \backslash sqrt\{L\; C\}\}$$

Rms current of a sinusoidal supply

$$I_{\text{rms}}=\frac{I_0}{\sqrt{2}}I\_\{\backslash text\{rms\}\}\; =\; \backslash frac\{I\_\{0\}\}\{\backslash sqrt\{2\}\}$$

Rms voltage of a sinusoidal supply

$$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}V\_\{\backslash text\{rms\}\}\; =\; \backslash frac\{V\_\{0\}\}\{\backslash sqrt\{2\}\}$$

Capacitors

Capacitance of parallel plates

$$C=\frac{{\epsilon }_{0}A}{d}C\; =\; \backslash frac\{\backslash varepsilon\_\{0\}\; A\}\{d\}$$

Charge on a capacitor

$$Q=CVQ\; =\; C\; V$$

Energy stored in a capacitor

$$\text{Ecap}=\frac{C{V}^2}{2}\backslash text\{Ecap\}\; =\; \backslash frac\{C\; V^\{2\}\}\{2\}$$

Circuits

Power from current and resistance

$$P=I^{2}RP\; =\; I^\{2\}\; R$$

Power from voltage and resistance

$$P=\frac{V^2}{R}P\; =\; \backslash frac\{V^\{2\}\}\{R\}$$

Resistance of a uniform wire

$$R=\frac{{\rho }_{e}L}{A}R\; =\; \backslash frac\{\backslash rho\_\{e\}\; L\}\{A\}$$

Two resistors in series and in parallel

$$r_s=r_1+r_2,r_p(r_1+r_2)=r_1{r}_{2}r\_\{s\}\; =\; r\_\{1\}\; +\; r\_\{2\},\backslash quad\; r\_\{p\}\; \backslash left(r\_\{1\}\; +\; r\_\{2\}\backslash right)\; =\; r\_\{1\}\; r\_\{2\}$$

Circular

Angular frequency and frequency

$$\omega =2\pi f\backslash omega\; =\; 2\; \backslash pi\; f$$

Centripetal acceleration

$$a=\frac{v^2}{r}a\; =\; \backslash frac\{v^\{2\}\}\{r\}$$

Centripetal force

$$F=\frac{m{v}^2}{r}F\; =\; \backslash frac\{m\; v^\{2\}\}\{r\}$$

Period and frequency

$$T=\frac{1}{f}T\; =\; \backslash frac\{1\}\{f\}$$

Speed on a circle from angular speed

$$v=\omega rv\; =\; \backslash omega\; r$$

Collisions

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

$$v_{1f}=\frac{(m_1-m_2)v_1+2m_2{v}_2}{m_1+m_2}v\_\{1f\}\; =\; \backslash frac\{\backslash left(m\_\{1\}\; -\; m\_\{2\}\backslash right)\; v\_\{1\}\; +\; 2\; m\_\{2\}\; v\_\{2\}\}\{m\_\{1\}\; +\; m\_\{2\}\}$$

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

$$v_{2f}=\frac{(m_2-m_1)v_2+2m_1{v}_1}{m_1+m_2}v\_\{2f\}\; =\; \backslash frac\{\backslash left(m\_\{2\}\; -\; m\_\{1\}\backslash right)\; v\_\{2\}\; +\; 2\; m\_\{1\}\; v\_\{1\}\}\{m\_\{1\}\; +\; m\_\{2\}\}$$

Dynamics

Acceleration down a frictionless incline

$$a=g\sin \left(\theta \right)a\; =\; g\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Average net force from a change of momentum

$$F=\frac{p_2-p_1}{t}F\; =\; \backslash frac\{p\_2\; -\; p\_1\}\{t\}$$

Component of the weight along an incline

$$F=mg\sin \left(\theta \right)F\; =\; m\; g\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Impulse of a constant force

$$J=FtJ\; =\; F\; t$$

Linear momentum

$$p=mvp\; =\; m\; v$$

Net force and acceleration

$$F=maF\; =\; m\; a$$

Net force with an opposing friction force

$$F=P-fF\; =\; P\; -\; f$$

Normal force on a level surface

$$N=WN\; =\; W$$

Normal force on an incline

$$N=mg\cos \left(\theta \right)N\; =\; m\; g\; \backslash cos\backslash left(\backslash theta\backslash right)$$

Sliding friction

$$f=\mu Nf\; =\; \backslash mu\; N$$

Tension in a rope holding a hanging body

$$T=mgT\; =\; m\; g$$

Weight in a uniform gravitational field

$$W=mgW\; =\; m\; g$$

Electricity

Charge carried by a steady current

$$Q=ItQ\; =\; I\; t$$

Electric power

$$P=VIP\; =\; V\; I$$

Voltage across an ohmic resistance

$$V=IRV\; =\; I\; R$$

Electrostatics

Electric field of a point charge

$$E_f=\frac{k_{e}Q}{r^2}E\_\{f\}\; =\; \backslash frac\{k\_\{e\}\; Q\}\{r^\{2\}\}$$

Electric potential of a point charge

$$V=\frac{k_{e}Q}{r}V\; =\; \backslash frac\{k\_\{e\}\; Q\}\{r\}$$

Force on a charge in an electric field

$$F=q{E}_{f}F\; =\; q\; E\_\{f\}$$

Potential difference in a uniform field

$$V=E_{f}dV\; =\; E\_\{f\}\; d$$

Energy

Average power

$$P=\frac{E}{t}P\; =\; \backslash frac\{E\}\{t\}$$

Energy stored in a linear spring

$$U=\frac{k{x}^2}{2}U\; =\; \backslash frac\{k\; x^\{2\}\}\{2\}$$

Force of a linear spring

$$F=kxF\; =\; k\; x$$

Gravitational potential energy near the surface

$$U=mghU\; =\; m\; g\; h$$

Power of a force on a moving body

$$P=FvP\; =\; F\; v$$

Speed of a body launched by a spring

$$\frac{k{x}^2}{2}=\frac{m{v}^2}{2}\backslash frac\{k\; x^\{2\}\}\{2\}\; =\; \backslash frac\{m\; v^\{2\}\}\{2\}$$

Translational kinetic energy

$$K=\frac{m{v}^2}{2}K\; =\; \backslash frac\{m\; v^\{2\}\}\{2\}$$

Work by a constant force

$$W=Fd\cos \left(\theta \right)W\; =\; F\; d\; \backslash cos\backslash left(\backslash theta\backslash right)$$

Work-energy theorem

$$W=\frac{m(v^2-v_0^2)}{2}W\; =\; \backslash frac\{m\; \backslash left(v^\{2\}\; -\; v\_0^\{2\}\backslash right)\}\{2\}$$

Fluids

Absolute pressure at a depth in a fluid

$$P=P_0+\rho ghP\; =\; P\_\{0\}\; +\; \backslash rho\; g\; h$$

Flow rate as volume per time

$$Q=\frac{V}{t}Q\; =\; \backslash frac\{V\}\{t\}$$

Flow rate through a cross section

$$Q=AvQ\; =\; A\; v$$

Mass density

$$\rho =\frac{m}{V}\backslash rho\; =\; \backslash frac\{m\}\{V\}$$

Pressure of a column of fluid

$$P=\rho ghP\; =\; \backslash rho\; g\; h$$

Pressure of a force on an area

$$P=\frac{F}{A}P\; =\; \backslash frac\{F\}\{A\}$$

Gravitation

Gravitational acceleration at a distance from a mass

$$g=\frac{GM}{r^2}g\; =\; \backslash frac\{G\; M\}\{r^\{2\}\}$$

Period of a circular orbit

$$T=2\pi \sqrt{\frac{r^3}{GM}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{r^\{3\}\}\{G\; M\}\}$$

Speed in a circular orbit

$$v=\sqrt{\frac{GM}{r}}v\; =\; \backslash sqrt\{\backslash frac\{G\; M\}\{r\}\}$$

Induction

Current dying away in an RL circuit

$$I=I_0{e}^{\frac{-t}{\tau }}I\; =\; I\_\{0\}\; e^\{\backslash frac\{-t\}\{\backslash tau\}\}$$

Emf of a rod moving through a magnetic field

$$\text{emf}=Blv\backslash text\{emf\}\; =\; B\; l\; v$$

Time constant of an RL circuit

$$\tau =\frac{L}{R}\backslash tau\; =\; \backslash frac\{L\}\{R\}$$

Kinematics

Displacement from the mean of initial and final velocity

$$x=\frac{(v_0+v)t}{2}x\; =\; \backslash frac\{\backslash left(v\_0\; +\; v\backslash right)\; t\}\{2\}$$

Displacement under constant acceleration

$$x=v_{0}t+\frac{a{t}^2}{2}x\; =\; v\_0\; t\; +\; \backslash frac\{a\; t^\{2\}\}\{2\}$$

Maximum height of a vertical throw

$$h=\frac{v_0^2}{2g}h\; =\; \backslash frac\{v\_0^\{2\}\}\{2\; g\}$$

Motion at constant velocity

$$d=vtd\; =\; v\; t$$

Speed after a displacement under constant acceleration

$$v^2=v_0^2+2axv^\{2\}\; =\; v\_0^\{2\}\; +\; 2\; a\; x$$

Speed after falling from rest

$$v=\sqrt{2gh}v\; =\; \backslash sqrt\{2\; g\; h\}$$

Time to the top of a vertical throw

$$t=\frac{v_0}{g}t\; =\; \backslash frac\{v\_0\}\{g\}$$

Velocity and displacement after constant acceleration

$$v=u+at,2s=(u+v)tv\; =\; u\; +\; a\; t,\backslash quad\; 2\; s\; =\; \backslash left(u\; +\; v\backslash right)\; t$$

Velocity under constant acceleration

$$v=v_0+atv\; =\; v\_0\; +\; a\; t$$

Magnetism

Magnetic field at the centre of a circular loop

$$B=\frac{{\mu }_{0}I}{2r}B\; =\; \backslash frac\{\backslash mu\_\{0\}\; I\}\{2\; r\}$$

Magnetic field inside a long solenoid

$$B={\mu }_0{n}_{t}IB\; =\; \backslash mu\_\{0\}\; n\_\{t\}\; I$$

Magnetic field of a long straight wire

$$B=\frac{{\mu }_{0}I}{2\pi r}B\; =\; \backslash frac\{\backslash mu\_\{0\}\; I\}\{2\; \backslash pi\; r\}$$

Magnetic force on a current-carrying wire

$$F=ILB\sin \left(\theta \right)F\; =\; I\; L\; B\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Magnetic force on a moving charge

$$F=qvB\sin \left(\theta \right)F\; =\; q\; v\; B\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Radius of a charged particle's circle in a magnetic field

$$r=\frac{mv}{qB\sin \left(\theta \right)}r\; =\; \backslash frac\{m\; v\}\{q\; B\; \backslash sin\backslash left(\backslash theta\backslash right)\}$$

Optics

Focal length of a spherical mirror

$$f=\frac{R}{2}f\; =\; \backslash frac\{R\}\{2\}$$

Index of refraction

$$n=\frac{c}{v}n\; =\; \backslash frac\{c\}\{v\}$$

Thin lens equation

$$\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}\backslash frac\{1\}\{d\_\{o\}\}\; +\; \backslash frac\{1\}\{d\_\{i\}\}\; =\; \backslash frac\{1\}\{f\}$$

Wavelength of light in a medium

$${\lambda }_n=\frac{\lambda }{n}\backslash lambda\_\{n\}\; =\; \backslash frac\{\backslash lambda\}\{n\}$$

Orbits

Escape speed

$$v=\sqrt{\frac{2GM}{r}}v\; =\; \backslash sqrt\{\backslash frac\{2\; G\; M\}\{r\}\}$$

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

$$T=2\pi \sqrt{\frac{a^3}{GM}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{a^\{3\}\}\{G\; M\}\}$$

Orbital speed on an elliptical orbit (vis-viva)

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

Oscillations

Amplitude-resonance frequency of a driven damped oscillator

$${\omega }_r=\sqrt{\frac{k}{m}-\frac{b^2}{2m^2}}\backslash omega\_\{r\}\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\; -\; \backslash frac\{b^\{2\}\}\{2\; m^\{2\}\}\}$$

Angular frequency of a damped oscillator

$${\omega }_d=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}\backslash omega\_\{d\}\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\; -\; \backslash frac\{b^\{2\}\}\{4\; m^\{2\}\}\}$$

Damping ratio

$$\zeta =\frac{b}{2\sqrt{km}}\backslash zeta\; =\; \backslash frac\{b\}\{2\; \backslash sqrt\{k\; m\}\}$$

Higher normal mode of a double pendulum

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

Higher normal-mode frequency of two coupled oscillators

$${\omega }_2=\sqrt{\frac{k+2k_c}{m}}\backslash omega\_\{2\}\; =\; \backslash sqrt\{\backslash frac\{k\; +\; 2\; k\_\{c\}\}\{m\}\}$$

Lower normal mode of a double pendulum

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

Maximum speed of a mass on a spring

$$v=X\sqrt{\frac{k}{m}}v\; =\; X\; \backslash sqrt\{\backslash frac\{k\}\{m\}\}$$

Period of a mass on a spring

$$T=2\pi \sqrt{\frac{m}{k}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{m\}\{k\}\}$$

Period of a physical pendulum

$$T=2\pi \sqrt{\frac{I_r}{mgd}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{I\_\{r\}\}\{m\; g\; d\}\}$$

Period of a simple pendulum

$$T=2\pi \sqrt{\frac{L_p}{g}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{L\_\{p\}\}\{g\}\}$$

Quality factor of a damped oscillator

$$Q_f=\frac{\sqrt{km}}{b}Q\_\{f\}\; =\; \backslash frac\{\backslash sqrt\{k\; m\}\}\{b\}$$

Steady-state amplitude of a driven damped oscillator

$$A=\frac{F_0}{\sqrt{{(k-m{\omega }^2)}^2+{\left(b\omega \right)}^2}}A\; =\; \backslash frac\{F\_\{0\}\}\{\backslash sqrt\{\backslash left(k\; -\; m\; \backslash omega^\{2\}\backslash right)^\{2\}\; +\; \backslash left(b\; \backslash omega\backslash right)^\{2\}\}\}$$

Quantum

De Broglie wavelength

$$\lambda =\frac{h}{p}\backslash lambda\; =\; \backslash frac\{h\}\{p\}$$

Energy of a photon

$$E=hfE\; =\; h\; f$$

Momentum of a photon

$$p=\frac{h}{\lambda }p\; =\; \backslash frac\{h\}\{\backslash lambda\}$$

Photon energy from wavelength

$$E=\frac{hc}{\lambda }E\; =\; \backslash frac\{h\; c\}\{\backslash lambda\}$$

RC circuits

Time constant of an RC circuit

$$\tau =RC\backslash tau\; =\; R\; C$$

Voltage of a charging capacitor

$$V=\text{emf}(1-e^{\frac{-t}{RC}})V\; =\; \backslash text\{emf\}\; \backslash left(1\; -\; e^\{\backslash frac\{-t\}\{R\; C\}\}\backslash right)$$

Voltage of a discharging capacitor

$$V=V_0{e}^{\frac{-t}{RC}}V\; =\; V\_\{0\}\; e^\{\backslash frac\{-t\}\{R\; C\}\}$$

Relativity

Rest energy

$$E_0=m{c}^{2}E\_\{0\}\; =\; m\; c^\{2\}$$

Rigid body

Acceleration of a body rolling without slipping down an incline

$$a=\frac{g\sin \left(\theta \right)}{1+\frac{I_r}{m{R}^2}}a\; =\; \backslash frac\{g\; \backslash sin\backslash left(\backslash theta\backslash right)\}\{1\; +\; \backslash frac\{I\_\{r\}\}\{m\; R^\{2\}\}\}$$

Acceleration of an Atwood machine with a massive pulley

$$a=\frac{(m_1-m_2)g}{m_1+m_2+\frac{I_r}{R^2}}a\; =\; \backslash frac\{\backslash left(m\_\{1\}\; -\; m\_\{2\}\backslash right)\; g\}\{m\_\{1\}\; +\; m\_\{2\}\; +\; \backslash frac\{I\_\{r\}\}\{R^\{2\}\}\}$$

Body-frame precession rate of a torque-free symmetric body

$${\mathrm{\Omega }}_p=\frac{{\omega }_3(I_3-I_1)}{I_1}\backslash Omega\_\{p\}\; =\; \backslash frac\{\backslash omega\_\{3\}\; \backslash left(I\_\{3\}\; -\; I\_\{1\}\backslash right)\}\{I\_\{1\}\}$$

Moment of inertia of a solid disk about its axis

$$I_r=\frac{m{R}^2}{2}I\_\{r\}\; =\; \backslash frac\{m\; R^\{2\}\}\{2\}$$

Moment of inertia of a solid sphere

$$I_r=\frac{2m{R}^2}{5}I\_\{r\}\; =\; \backslash frac\{2\; m\; R^\{2\}\}\{5\}$$

Moment of inertia of a thin rod about its center

$$I_r=\frac{m{L}_r^2}{12}I\_\{r\}\; =\; \backslash frac\{m\; L\_\{r\}^\{2\}\}\{12\}$$

Moment of inertia of a thin rod about one end

$$I_r=\frac{m{L}_r^2}{3}I\_\{r\}\; =\; \backslash frac\{m\; L\_\{r\}^\{2\}\}\{3\}$$

Moment of inertia of a thin spherical shell

$$I_r=\frac{2m{R}^2}{3}I\_\{r\}\; =\; \backslash frac\{2\; m\; R^\{2\}\}\{3\}$$

Rotating frames

Off-axis equilibrium of a bead on a rotating hoop

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

Precession period of a Foucault pendulum

$$T=\frac{2\pi }{\mathrm{\Omega }\sin \left(\varphi \right)}T\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash Omega\; \backslash sin\backslash left(\backslash phi\backslash right)\}$$

Rotation

Angular momentum of a point mass on a circle

$$L=mvrL\; =\; m\; v\; r$$

Angular momentum of a rotating body

$$L=I_r\omega L\; =\; I\_\{r\}\; \backslash omega$$

Angular velocity and angle turned under constant angular acceleration

$${\omega }^2={\omega }_0^2+2\alpha \theta \backslash omega^\{2\}\; =\; \backslash omega\_\{0\}^\{2\}\; +\; 2\; \backslash alpha\; \backslash theta$$

Angular velocity under constant angular acceleration

$$\omega ={\omega }_0+\alpha t\backslash omega\; =\; \backslash omega\_\{0\}\; +\; \backslash alpha\; t$$

Centripetal acceleration from angular velocity

$$a=r{\omega }^{2}a\; =\; r\; \backslash omega^\{2\}$$

Rotational kinetic energy

$$K_R=\frac{I_r{\omega }^2}{2}K\_\{R\}\; =\; \backslash frac\{I\_\{r\}\; \backslash omega^\{2\}\}\{2\}$$

Tangential acceleration of a point on a rotating body

$$a=r\alpha a\; =\; r\; \backslash alpha$$

Torque of a force

$$\tau =rF\sin \left(\theta \right)\backslash tau\; =\; r\; F\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Thermal

Carnot (maximum) efficiency

$$\text{eff}=1-\frac{T_c}{T_h}\backslash text\{eff\}\; =\; 1\; -\; \backslash frac\{T\_\{c\}\}\{T\_\{h\}\}$$

Coefficient of performance of a heat pump

$$\text{cop}=\frac{Q_h}{W}\backslash text\{cop\}\; =\; \backslash frac\{Q\_\{h\}\}\{W\}$$

Coefficient of performance of a refrigerator

$$\text{cop}=\frac{Q_c}{W}\backslash text\{cop\}\; =\; \backslash frac\{Q\_\{c\}\}\{W\}$$

Efficiency of a heat engine

$$\text{eff}=\frac{W}{Q_h}\backslash text\{eff\}\; =\; \backslash frac\{W\}\{Q\_\{h\}\}$$

Energy balance of a cyclic heat engine

$$W=Q_h-Q_{c}W\; =\; Q\_\{h\}\; -\; Q\_\{c\}$$

Heat for a temperature change

$$Q=mc\mathrm{\Delta }TQ\; =\; m\; c\; \backslash Delta\; T$$

Heat to boil or condense

$$Q=m{L}_{v}Q\; =\; m\; L\_\{v\}$$

Heat to melt or freeze

$$Q=m{L}_{f}Q\; =\; m\; L\_\{f\}$$

Ideal gas law

$$PV=nRTP\; V\; =\; n\; R\; T$$

Linear thermal expansion

$$d_L={\alpha }_{t}L{d}_{T}d\_\{L\}\; =\; \backslash alpha\_\{t\}\; L\; d\_\{T\}$$

Thermal expansion of an area

$$d_A=2{\alpha }_{t}A{d}_{T}d\_\{A\}\; =\; 2\; \backslash alpha\_\{t\}\; A\; d\_\{T\}$$

Thermal expansion of volume

$$d_V=\beta V_0{d}_{T}d\_\{V\}\; =\; \backslash beta\; V\_\{0\}\; d\_\{T\}$$

Variable mass

Velocity change of a rocket (Tsiolkovsky)

$$\mathrm{\Delta }v=v_e\ln \left(\frac{m_0}{m_f}\right)\backslash Delta\; v\; =\; v\_\{e\}\; \backslash ln\backslash left(\backslash frac\{m\_\{0\}\}\{m\_\{f\}\}\backslash right)$$

Waves

Intensity as power per area

$$I=\frac{P}{A}I\; =\; \backslash frac\{P\}\{A\}$$

Speed of a periodic wave

$$v=f\lambda v\; =\; f\; \backslash lambda$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified classical physics capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (kinematics, dynamics, energy, momentum, rotation, gravitation, oscillations, waves and sound, optics, thermodynamics, electricity and magnetism); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{classical_physics_ai999x_1_0_0,
  title  = {classical-physics-ai999x: an offline, verified classical physics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/classical-physics-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/classical-physics-ai999x/discussions or visit https://ai999x.com.

Support the project

classical-physics-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com