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50 formulas

Mechanics

Forces, motion, energy. Every formula below opens into a live, hands-on simulation.

dynamics
F=maF = ma

Newton's Second Law

Force equals mass times acceleration: heavier objects need more push.

elasticity
F=−kxF = -kx

Hooke's Law

A spring pushes back proportionally to how far you stretch it.

gravitation
F=GMmr2F = \frac{GMm}{r^2}

Newton's Law of Universal Gravitation

Every mass attracts every other mass; force drops with the square of distance.

energy
KE=12mv2KE = \frac{1}{2}mv^2

Kinetic Energy

Energy of motion: doubles with mass, quadruples with speed.

circular motion
ac=v2ra_c = \frac{v^2}{r}

Centripetal Acceleration

Moving in a circle requires constant inward acceleration; faster or tighter = more.

kinematics
R=v2sin⁡(2θ)gR = \frac{v^2 \sin(2\theta)}{g}

Projectile Range

Launch angle of 45° maximizes range; steeper or flatter angles fall shorter.

energy
Wnet=ΔKE=12mvf2−12mvi2W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2

Work-Energy Theorem

Net work on an object equals the change in its kinetic energy.

oscillations
T=2πmkT = 2\pi\sqrt{\frac{m}{k}}

Simple Harmonic Motion Period

Heavier masses oscillate slower; stiffer springs oscillate faster.

dynamics
p=mvp = mv

Linear Momentum

Momentum measures how hard it is to stop a moving object — it scales with both mass and speed.

dynamics
J=FΔt=ΔpJ = F \Delta t = \Delta p

Impulse-Momentum Theorem

Stopping the same momentum over a longer time means smaller force — the whole point of airbags, crumple zones, and bending your knees.

energy
U=mghU = mgh

Gravitational Potential Energy

Lifting something stores energy in the gravitational field — drop it and the energy reappears as motion.

energy
P=Wt=FvP = \frac{W}{t} = Fv

Mechanical Power

Power is how fast you spend energy. Same work, less time = more power.

rotation
τ=rFsin⁡θ\tau = rF\sin\theta

Torque

Torque is twist. The farther you push from the pivot, and the more perpendicular your push, the more spin you create.

rotation
L=IωL = I\omega

Angular Momentum

Angular momentum is rotational inertia times spin speed. Squeeze inward and you must spin faster to keep it conserved.

rotation
KErot=12Iω2KE_{\text{rot}} = \tfrac{1}{2} I \omega^2

Rotational Kinetic Energy

Spinning things store energy just like moving things — and a small radius increase pays off big because I scales with r².

kinematics
v=2ghv = \sqrt{2gh}

Free Fall Velocity

Drop something from height h in vacuum — by the time it hits, all the gravitational potential mgh has turned into kinetic ½mv².

oscillations
T=2πLgT = 2\pi \sqrt{\frac{L}{g}}

Simple Pendulum Period

Longer pendulums swing slower. The period depends only on length and gravity — not mass and (almost) not on amplitude.

rotational dynamics
I=mr2I = mr^2

Moment of Inertia (Point Mass)

Mass placed far from the rotation axis resists rotation much more than the same mass placed close in.

gravitation
vesc=2GMRv_{\text{esc}} = \sqrt{\frac{2GM}{R}}

Escape Velocity

The speed at which kinetic energy exactly equals the gravitational binding energy — no faster, you fly free.

gravitation
T2=4π2GMa3T^2 = \frac{4\pi^2}{GM} a^3

Kepler's Third Law

Farther orbits take longer — and not linearly: the period grows as the 3/2 power of the orbital radius.

dynamics
Fk=μkNF_k = \mu_k N

Kinetic Friction Force

Once an object is already sliding, friction resists motion with a force proportional to how hard the surfaces are pressed together.

dynamics
Fs,max⁡=μsNF_{s,\max} = \mu_s N

Maximum Static Friction

Static friction adjusts itself to match whatever you push with — up to a hard ceiling set by μ_s N. Cross that line and it gives way.

dynamics
N=mgcos⁡θN = m g \cos\theta

Normal Force on an Incline

The surface only needs to support the part of gravity pointing into it — cos θ of the weight.

dynamics
F∥=mgsin⁡θF_\parallel = m g \sin\theta

Gravity Component Along an Incline

Only the sine-component of gravity accelerates an object down the slope — the cosine-component is canceled by the normal force.

dynamics
T=2m1m2gm1+m2T = \frac{2 m_1 m_2 g}{m_1 + m_2}

Atwood Machine Tension

The tension is twice the harmonic mean of the two weights — equal masses give T = mg (no motion), unequal masses give something in between.

dynamics
v1′=(m1−m2)v1+2m2v2m1+m2v_1' = \frac{(m_1 - m_2) v_1 + 2 m_2 v_2}{m_1 + m_2}

1D Elastic Collision (Final Velocities)

Conserve both momentum and kinetic energy and the velocities of the two bodies swap (when masses are equal) or rearrange linearly in mass ratios.

dynamics
vf=m1v1+m2v2m1+m2v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}

Perfectly Inelastic Collision (1D)

Momentum is conserved, but kinetic energy is not — the bodies stick together and share a single final velocity equal to the system center-of-mass velocity.

dynamics
e=−v1′−v2′v1−v2e = -\frac{v_1' - v_2'}{v_1 - v_2}

Coefficient of Restitution

A single dimensionless number, 0 ≤ e ≤ 1, that says how much of the relative speed survives a collision. e=1 is perfectly elastic; e=0 is perfectly inelastic.

dynamics
xcm=m1x1+m2x2m1+m2x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}

Center of Mass (Two Particles)

A mass-weighted average of positions — the single point that behaves like the whole system for Newton's second law.

rotational dynamics
I=Icm+md2I = I_{cm} + m d^2

Parallel Axis Theorem

Spinning about a shifted axis costs extra rotational inertia equal to m·d² — because every particle now traces a larger circle.

rotational dynamics
v=Rωv = R \omega

Rolling Without Slipping

The contact point of a rolling wheel is instantaneously at rest — so the translation speed of the center equals the tangential speed at the rim.

rotational dynamics
I1ω1=I2ω2I_1 \omega_1 = I_2 \omega_2

Conservation of Angular Momentum

With no external torque, the product Iω is fixed: pull mass closer to the axis and you must spin faster to compensate.

analytical mechanics
S[q]=∫t1t2L(q,q˙,t) dt,δS=0S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t)\, dt, \qquad \delta S = 0

Principle of Stationary Action

Of all imaginable paths between two events, nature takes the one whose action doesn't change under small wiggles.

analytical mechanics
ddt(∂L∂q˙i)−∂L∂qi=0\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0

Euler–Lagrange Equation

Pick any coordinates you like, write energy in, equations of motion come out — no force diagrams needed.

analytical mechanics
pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}

Generalized (Conjugate) Momentum

Each coordinate gets its own momentum — differentiate L by the velocity of that coordinate.

analytical mechanics
H(q,p,t)=∑ipiq˙i−LH(q, p, t) = \sum_i p_i \dot{q}_i - L

The Hamiltonian (Legendre Transform)

Swap each velocity for its momentum; what's left is (usually) the total energy as a function of position and momentum.

analytical mechanics
q˙i=∂H∂pi,p˙i=−∂H∂qi\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}

Hamilton's Canonical Equations

H is a landscape over phase space; states flow along its contour lines, position fed by momentum and momentum drained by force.

analytical mechanics
{f,g}=∑i(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi)\{f, g\} = \sum_i \left( \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i} \right)

Poisson Bracket & Liouville Flow

The bracket measures how two phase-space functions 'stir' each other; pairing anything with H gives its time evolution.

analytical mechanics
∂L∂qi=0  ⇒  ddt∂L∂q˙i=0\frac{\partial L}{\partial q_i} = 0 \;\Rightarrow\; \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} = 0

Noether's Theorem

Every continuous symmetry of the action hands you a conserved quantity — no exceptions, no extra work.

analytical mechanics
L=12(m1+m2)l12θ˙12+12m2l22θ˙22+m2l1l2θ˙1θ˙2cos⁡(θ1−θ2)+(m1+m2)gl1cos⁡θ1+m2gl2cos⁡θ2\mathcal{L} = \tfrac{1}{2}(m_1{+}m_2)l_1^2\dot\theta_1^2 + \tfrac{1}{2}m_2 l_2^2\dot\theta_2^2 + m_2 l_1 l_2 \dot\theta_1\dot\theta_2\cos(\theta_1{-}\theta_2) + (m_1{+}m_2)gl_1\cos\theta_1 + m_2 gl_2\cos\theta_2

Double Pendulum (Lagrangian Chaos)

Two pendulums chained together: the Lagrangian is easy to write, the motion is impossible to predict long-term.

gravitation
Ftidal≈2GMmRd3F_{\text{tidal}} \approx \frac{2GMmR}{d^{3}}

Tidal Force

Gravity weakens with distance, so the Moon pulls Earth's near side harder than its center, and the center harder than the far side. In Earth's frame that difference stretches the planet along the Moon line and squeezes it sideways — two ocean bulges, two high tides a day. Because the effect goes as 1/d³ (not 1/d²), the nearby Moon out-tides the enormous Sun.

collisions
∑imiv⃗i=constant⟺m1u⃗1+m2u⃗2=m1v⃗1+m2v⃗2\sum_i m_i \vec{v}_i = \text{constant} \quad\Longleftrightarrow\quad m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2

Conservation of Linear Momentum

With no outside push, the total momentum of a system can't change — collisions only trade it between parts.

dynamics with drag
vt=2mgρ Cd Av_t = \sqrt{\dfrac{2 m g}{\rho\, C_d\, A}}

Terminal Velocity

You stop accelerating when quadratic air drag grows to exactly match your weight.

rotating frames
a⃗Cor=−2 ω⃗×v⃗\vec{a}_{\text{Cor}} = -2\,\vec{\omega}\times\vec{v}

Coriolis Deflection

In a rotating frame, anything that moves gets pushed sideways — the frame, not a real force, does the deflecting.

rigid body dynamics
τ⃗=dL⃗dt,Ωp=τL=τIω\vec{\tau} = \frac{d\vec{L}}{dt}, \qquad \Omega_p = \frac{\tau}{L} = \frac{\tau}{I\omega}

Gyroscopic Precession

A torque changes angular momentum's direction, not the spin's size, so the axle swings sideways instead of falling.

rigid body dynamics
L⃗=Ifront ω⃗front+Iback ω⃗back=0\vec{L} = I_{\text{front}}\,\vec{\omega}_{\text{front}} + I_{\text{back}}\,\vec{\omega}_{\text{back}} = 0

The Falling-Cat Problem

Change your shape in a cycle and you can reorient with zero angular momentum — geometry rotates you, not spin.

ballistics
R=v2sin⁡2θg,v=2 η MghmR = \frac{v^2 \sin 2\theta}{g}, \qquad v = \sqrt{\frac{2\,\eta\, M g h}{m}}

Physics of the Trebuchet

A heavy counterweight's drop becomes a light stone's speed; range then follows the projectile formula.

oscillations instability
θ¨+2ζnet ωn θ˙+ωn2 θ=0,ζnet=ζstruct−ζaero(U)\ddot{\theta} + 2\zeta_{\text{net}}\,\omega_n\,\dot{\theta} + \omega_n^2\,\theta = 0, \qquad \zeta_{\text{net}} = \zeta_{\text{struct}} - \zeta_{\text{aero}}(U)

Tacoma Narrows: Aeroelastic Flutter

Past a critical wind, the airflow pumps energy in faster than damping drains it, so oscillations blow up.

celestial mechanics
Δϕ=6πGMa (1−e2) c2per orbit\Delta\phi = \frac{6\pi G M}{a\,(1-e^2)\,c^2} \quad \text{per orbit}

Perihelion Precession of Mercury

Relativity bends spacetime just enough that the orbit's closest point creeps forward each lap.

astrophysics
MCh≈ω30 3π2(ℏcG)3/21(μemH)2≈1.44(2μe)2M⊙M_{\text{Ch}} \approx \frac{\omega_3^0\,\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2} \approx 1.44\left(\frac{2}{\mu_e}\right)^2 M_\odot

The Chandrasekhar Limit

Relativistic electron pressure scales like gravity, so above ~1.4 solar masses gravity always wins.