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Work is done when a force moves an object through a displacement in the direction of the force. The formula is W = F·d·cos(θ), where F is the applied force, d is the displacement, and θ is the angle between the force vector and displacement vector. If force and displacement are parallel (θ = 0°), W = Fd. If they are perpendicular (θ = 90°), no work is done — which is why carrying a heavy bag horizontally requires no mechanical work despite the effort felt.
Kinetic energy (KE) is the energy of motion: KE = ½mv², where m is mass and v is speed. Potential energy (PE) is stored energy due to position or configuration. Gravitational PE = mgh, where h is the height above a reference level. Both are measured in Joules (J): 1 J = 1 N·m = 1 kg·m²/s².
The Work-Energy Theorem is a powerful result: the net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv² − ½mu². This connects the dynamics (forces) to the kinematics (velocities) without needing to track time.
Worked example: A 2 kg ball is dropped from h = 5 m. PE at top = 2×9.81×5 = 98.1 J. By conservation of energy, KE at bottom = 98.1 J. So ½×2×v² = 98.1, giving v = √98.1 ≈ 9.9 m/s.
W = F·d·cos(θ) | KE = ½mv² | PE = mgh
W = F × d × cos(θ)
θ = angle between force direction and displacement. Use 0° for force parallel to motion.
Energy Formulas Reference
| Formula | Type | Variables |
|---|---|---|
| W = Fd cos(θ) | Work | F = force, d = displacement, θ = angle |
| KE = ½mv² | Kinetic Energy | m = mass, v = velocity |
| PE = mgh | Gravitational PE | m = mass, g = 9.81, h = height |
| W_net = ΔKE | Work-Energy Theorem | Change in kinetic energy |
| E_total = KE + PE | Mechanical Energy | Conserved (no friction) |
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.