Introduction
Welcome to the most comprehensive Mechanics Formulas Guide. Classical mechanics is the branch of physics that deals with the motion of objects and the forces that cause or change that motion. From the falling of an apple to the orbit of planets, mechanics provides the mathematical framework to understand and predict the physical world around us.
Whether you're a high school student preparing for exams, an engineering student tackling dynamics, or a curious mind exploring physics, this guide will give you a complete reference of the essential mechanics formulas, with clear explanations and worked examples.
This guide covers kinematics, Newton's three laws of motion, work-energy-power relationships, momentum and collisions, circular motion, rotational dynamics, universal gravitation, simple harmonic motion, common mistakes to avoid, and real-world worked examples.
What is Classical Mechanics?
Classical mechanics (also called Newtonian mechanics) is the branch of physics developed by Sir Isaac Newton in the late 17th century. It describes the motion of macroscopic objects—from projectiles to planets to galaxies—under the influence of forces.
Three Main Branches
Kinematics
Describes motion without considering the forces that cause it (position, velocity, acceleration).
Dynamics
Studies the forces and torques that cause or change motion (Newton's laws).
Statics
Studies objects at rest or in equilibrium (net force = 0).
Classical mechanics works for everyday objects moving at speeds much less than the speed of light. For objects near light speed, use Special Relativity. For atomic-scale objects, use Quantum Mechanics. For very strong gravitational fields, use General Relativity.
Kinematics (Motion)
Kinematics describes motion using variables like displacement, velocity, acceleration, and time—without worrying about the forces involved.
The 4 Kinematic Equations (Constant Acceleration)
| Equation | Name | Use When You Know |
|---|---|---|
| v = v₀ + at | Velocity-Time | v₀, a, t → find v |
| x = v₀t + ½at² | Displacement-Time | v₀, a, t → find x |
| v² = v₀² + 2ax | Velocity-Displacement | v₀, a, x → find v |
| x = ½(v₀ + v)t | Average Velocity | v₀, v, t → find x |
a = acceleration (m/s²) • t = time (s)
Free Fall (Vertical Motion Under Gravity)
When an object falls freely near Earth's surface, it experiences a constant downward acceleration of g ≈ 9.81 m/s². Simply replace 'a' with 'g' (or '-g' if upward is positive) in the kinematic equations.
Newton's Laws of Motion
Sir Isaac Newton formulated three fundamental laws in 1687 that form the foundation of classical mechanics.
Common Forces
| Force | Formula | Description |
|---|---|---|
| Weight | W = mg | Gravitational force on an object |
| Normal Force | N | Perpendicular contact force from a surface |
| Friction (kinetic) | f = μₖN | Opposes sliding motion |
| Friction (static, max) | f ≤ μₛN | Prevents motion from starting |
| Spring Force | F = -kx | Hooke's Law (restoring force) |
| Tension | T | Pulling force through a string/rope |
Work, Energy & Power
Energy is the capacity to do work. The work-energy theorem connects force, motion, and energy in a powerful way.
Types of Energy
| Energy Type | Formula | Description |
|---|---|---|
| Kinetic Energy | KE = ½mv² | Energy of motion |
| Gravitational PE | PE = mgh | Energy due to height |
| Elastic PE | PE = ½kx² | Energy stored in a spring |
| Power | P = W/t = Fv | Rate of doing work |
The Work-Energy Theorem
If only conservative forces: KEi + PEi = KEf + PEf
Energy cannot be created or destroyed—only transformed. In a closed system with no friction, total mechanical energy (KE + PE) remains constant. This principle is one of the most powerful tools in physics.
Momentum & Collisions
Momentum is a measure of an object's "quantity of motion" and is conserved in all collisions (when no external forces act).
Impulse-Momentum Theorem
Types of Collisions
| Collision Type | Momentum | Kinetic Energy | Example |
|---|---|---|---|
| Elastic | Conserved ✓ | Conserved ✓ | Billiard balls |
| Inelastic | Conserved ✓ | Lost (→ heat) | Car crash |
| Perfectly Inelastic | Conserved ✓ | Max lost | Clay sticking together |
Circular Motion
When an object moves in a circle, it experiences centripetal acceleration directed toward the center, even if its speed is constant.
Key Formulas
| Quantity | Formula | Unit |
|---|---|---|
| Centripetal Acceleration | ac = v²/r = ω²r | m/s² |
| Centripetal Force | Fc = mv²/r = mω²r | N |
| Angular Velocity | ω = 2π/T = v/r | rad/s |
| Period | T = 2πr/v = 2π/ω | s |
| Frequency | f = 1/T = ω/(2π) | Hz |
Centripetal force is a real force directed toward the center (e.g., tension, gravity, friction). Centrifugal force is a "fictitious" force experienced in a rotating reference frame—it's not a real force, just the effect of inertia.
Rotational Motion
Rotational motion is the angular analog of linear motion. Every linear quantity has a rotational counterpart.
Linear vs. Rotational Analogs
| Linear | Rotational | Relationship |
|---|---|---|
| Displacement (x) | Angular displacement (θ) | x = rθ |
| Velocity (v) | Angular velocity (ω) | v = rω |
| Acceleration (a) | Angular acceleration (α) | a = rα |
| Mass (m) | Moment of inertia (I) | Resistance to rotation |
| Force (F) | Torque (τ) | τ = r × F |
| Momentum (p) | Angular momentum (L) | L = Iω |
Newton's Second Law for Rotation
Moments of Inertia (Common Shapes)
| Shape | Moment of Inertia (I) | Axis |
|---|---|---|
| Solid cylinder/disk | ½MR² | Central axis |
| Solid sphere | (2/5)MR² | Through center |
| Thin rod | (1/12)ML² | Through center |
| Hoop/ring | MR² | Central axis |
Gravitation
Newton's Law of Universal Gravitation describes the attractive force between any two masses in the universe.
Key Constants & Values
| Quantity | Symbol | Value |
|---|---|---|
| Gravitational Constant | G | 6.674 × 10⁻¹¹ N·m²/kg² |
| Mass of Earth | ME | 5.972 × 10²⁴ kg |
| Radius of Earth | RE | 6.371 × 10⁶ m |
| Surface Gravity | g | 9.81 m/s² |
Orbital Mechanics
Simple Harmonic Motion
Simple Harmonic Motion (SHM) is periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.
Key Formulas
| Quantity | Formula | Description |
|---|---|---|
| Displacement | x(t) = A cos(ωt + φ) | Position at time t |
| Velocity | v(t) = -Aω sin(ωt + φ) | Rate of change of position |
| Acceleration | a(t) = -Aω² cos(ωt + φ) | Rate of change of velocity |
| Angular Frequency | ω = 2πf = 2π/T | rad/s |
| Period (mass-spring) | T = 2π√(m/k) | Independent of amplitude |
| Period (pendulum) | T = 2π√(L/g) | Small angles only |
Total mechanical energy is conserved in SHM: E = ½kA² = ½mv² + ½kx². Energy continuously transforms between kinetic and potential forms.
Common Mistakes
Mechanics is full of subtle traps. Avoid these common pitfalls.
Mass vs. Weight
Using "kg" for weight or "N" for mass.
Sign Convention
Forgetting negative signs for downward motion or restoring forces.
Units Mismatch
Mixing km/h with m/s, or grams with kilograms.
Confusing v and v²
Using v² = v₀² + 2ax when you meant v = v₀ + at.
Friction Direction
Forgetting friction always opposes relative motion.
Angle Units
Using degrees in formulas that require radians (ω, θ).
Worked Examples
Let's apply these formulas to real-world problems.
Example 1: Kinematics (Free Fall)
Problem: A ball is dropped from a height of 45 m. How long does it take to hit the ground? What is its final velocity?
Example 2: Newton's Second Law
Problem: A 1200 kg car accelerates from 0 to 25 m/s in 8 seconds. What is the net force required?
Example 3: Conservation of Energy
Problem: A 2 kg ball is dropped from 10 m. What is its speed just before hitting the ground? (Ignore air resistance)
Example 4: Momentum Conservation
Problem: A 1000 kg car moving at 20 m/s collides with a stationary 1500 kg truck. They stick together. What is their final velocity?
Example 5: Centripetal Force
Problem: A 0.5 kg ball is whirled in a horizontal circle of radius 2 m at 4 m/s. What is the tension in the string?
Example 6: Simple Pendulum
Problem: What is the period of a pendulum with length 1 m on Earth?
Follow this 5-step method: (1) Draw a diagram, (2) List knowns and unknowns, (3) Choose the right formula, (4) Solve algebraically first, (5) Plug in numbers and check units.
Conclusion
Classical mechanics is one of humanity's greatest intellectual achievements. From Newton's laws to Einstein's relativity, the formulas in this guide form the foundation of physics, engineering, and our understanding of the physical universe. Master these formulas, and you'll have the tools to analyze everything from a falling apple to a rocket launching to orbit.
Key Takeaways
- 4 Kinematic Equations: Cover all constant-acceleration motion problems.
- F = ma: The most important equation in classical mechanics.
- Energy Conservation: A powerful shortcut when forces are complex.
- Momentum Conservation: Essential for analyzing collisions.
- Centripetal Force: Required for any circular motion.
- Universal Gravitation: Explains orbits, tides, and planetary motion.
- SHM: Describes springs, pendulums, and countless oscillating systems.
If I have seen further, it is by standing on the shoulders of giants. The laws of mechanics are the foundation upon which all of modern physics and engineering are built.
Ready to apply these formulas? Explore our suite of free physics calculators below to solve real-world mechanics problems instantly. From projectile motion to gravitational forces, we've got you covered. Happy calculating! ⚙️🚀✨