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Classical Mechanics: Formula Cheat Sheet
Every key formula for Classical Mechanics, with what it means and where it comes from.
| Formula | What it means | Topic |
|---|---|---|
| a = v^2 / r | Centripetal acceleration equals speed squared divided by radius. | Circular Motion and Centripetal Force |
| F = m v^2 / r | Centripetal force equals mass times speed squared divided by radius. | Circular Motion and Centripetal Force |
| p = m v | Momentum equals mass times velocity. | Conservation of Momentum |
| m1 u1 + m2 u2 = m1 v1 + m2 v2 | Total momentum before a collision equals total momentum after it. | Conservation of Momentum |
| F = μ N | Frictional force equals the coefficient of friction times the normal force. | Friction: Static, Kinetic and Rolling |
| F_s,max = μ_s N | Maximum static friction before an object starts to slide. | Friction: Static, Kinetic and Rolling |
| F_k = μ_k N | Kinetic friction while an object is sliding. | Friction: Static, Kinetic and Rolling |
| F_net = 0, so a = 0 | When the net force is zero, acceleration is zero and velocity does not change. | Newton's First Law of Motion |
| F = m a | Net force equals mass times acceleration. | Newton's Second Law of Motion |
| a = F / m | Acceleration equals net force divided by mass. | Newton's Second Law of Motion |
| 1 N = 1 kg m/s^2 | One newton accelerates a 1 kg mass at 1 m/s². | Newton's Second Law of Motion |
| F_AB = -F_BA | The force A exerts on B is equal and opposite to the force B exerts on A. | Newton's Third Law of Motion |
| W = F d | Work equals force times distance moved in the direction of the force. | Work and the Work-Energy Theorem |
| KE = 1/2 m v^2 | Kinetic energy equals half the mass times the speed squared. | Work and the Work-Energy Theorem |
| W_net = 1/2 m v^2 - 1/2 m u^2 | Net work equals the change in kinetic energy. | Work and the Work-Energy Theorem |
| PE = m g h | Gravitational potential energy equals mass times g times height. | Work and the Work-Energy Theorem |
| tan θ = v² ÷ (r × g) | On a banked curve the road's slope supplies the sideways force, so the ideal angle depends on speed and bend radius. | Banked Curve Calculator |
| F_b = ρ × V × g | The buoyant force equals the weight of the fluid pushed aside, so denser fluids and larger volumes lift more. | Buoyancy Calculator |
| F_c = m × v² ÷ r | The force pulling an object around a circle grows with mass and speed squared, and shrinks with a larger radius. | Centripetal Force Calculator |
| a = v² ÷ r | An object circling at steady speed still accelerates toward the centre, with the size set by speed squared over radius. | Circular Motion Calculator |
| v = √(2 × G × M ÷ r) | Escape speed depends only on the body's mass and radius, not on the mass of the object trying to escape. | Escape Velocity Calculator |
| g = G × M ÷ r² | Gravitational field strength falls with the square of distance from the centre of the mass that creates it. | Gravitational Field Calculator |
| PE = m × g × h | Lifting an object stores energy equal to its weight times the height it is raised, measured from a chosen zero level. | Gravitational Potential Energy Calculator |
| F = k × x | Within its elastic limit a spring's restoring force is directly proportional to how far it is stretched from rest. | Hooke's Law Calculator |
| J = F × Δt = Δp | Impulse is force times the time it acts, and it equals the change in momentum of the object it pushes. | Impulse and Momentum Calculator |
| KE = ½ × m × v² | Kinetic energy grows with mass and with the square of speed, so doubling speed quadruples the energy. | Kinetic Energy Calculator |
| p = m × v | Momentum is mass times velocity, so a heavy slow object can carry the same momentum as a light fast one. | Momentum Calculator |
| F_net = F₁ + F₂ | Forces acting in the same direction add together to give the net force, while a force in the opposite direction subtracts from it. | Net Force Calculator |
| F = m × a | The force on an object equals its mass times its acceleration, so the same force accelerates a lighter object more. | Newton's Second Law Calculator |
| T = 2π × √(L ÷ g) | A pendulum's period depends on its length and gravity only, not on the bob's mass or the size of a small swing. | Pendulum Period Calculator |
| τ = r × F × sin θ | Torque is the turning effect of a force, largest when the push is at right angles to the lever arm. | Torque Calculator |
| R = √(A_x² + A_y²) | A vector's size comes from its two perpendicular components by Pythagoras, and its angle can be negative below the axis. | Vector Addition Calculator |
| W = F × d × cos θ | Only the part of a force along the direction of motion does work, and power is simply how fast that work is done. | Work and Power Calculator |
| E = F × L ÷ (A × ΔL) | Young's modulus measures stiffness: a stiffer material stretches less under the same pull for the same size of sample. | Young's Modulus Calculator |