Work done by a constant force. θ is the angle between F and d. Cos 90° = 0 → perpendicular forces do zero work.
W_net = ΔK
Work-energy theorem. Net work = change in kinetic energy. Use it instead of F = ma + kinematics.
U_g = mgh
Gravitational PE near Earth. h is the height above your chosen reference (zero) level.
U_g = −Gm₁m₂/r
Universal gravitational PE. For planets/satellites. Negative; approaches 0 as r → ∞.
U_s = ½kx²
Elastic PE in a spring. x is displacement from equilibrium. Always positive.
K_i + U_i = K_f + U_f
Conservation of mechanical energy. Use when ONLY conservative forces (gravity, springs) do work.
E_dissipated = F_f · d
Energy lost to friction. Friction force times path length. Becomes thermal energy.
P = ΔE/Δt
Average power. Energy transferred divided by time. SI unit: watt (W).
P = F·v
Instantaneous power. Useful when force and velocity are along the same direction.
The 5 topics at a glance
3.1 Translational Kinetic Energy
K = ½mv². Energy of motion of an object's center of mass. Scalar; always ≥ 0. Reference-frame dependent — different observers can measure different KE values.
3.2 Work
W = F·d·cos(θ). Energy transferred by a force over a distance. Positive if F aligns with d; negative if it opposes; zero if perpendicular. Work-energy theorem: W_net = ΔK.
3.3 Potential Energy
Stored energy from position/configuration. U_g = mgh (near Earth), U_g = −Gm₁m₂/r (universal), U_s = ½kx² (spring). Only conservative forces have associated PE.
3.4 Conservation of Energy
Energy is never created or destroyed. ME = K + U is conserved when only conservative forces act. Friction dissipates ME into thermal energy.
3.5 Power
P = ΔE/Δt. The rate of energy transfer. Units: watts (J/s). For constant F along v: P = F·v. A 100 W lightbulb uses 100 J every second.
Connecting It All
Energy methods are often MUCH faster than F = ma. Look for problems involving height changes, springs, or "find the final speed" — those scream energy conservation.
🧠 How to solve an energy problem (5 steps)
1. Choose your system. What's included? (The block? The block + Earth? The block + spring?)
2. Identify initial and final states. Where does the motion start? Where does it end?
3. Pick a zero of potential energy. The lowest point is often easiest.
4. Check which forces act. Only conservative? → ME is conserved. Friction or drag? → some ME dissipated.
5. Write the energy equation and solve. Usually K_i + U_i = K_f + U_f (with an extra "lost to friction" term if needed).
⚡ When to use energy vs. forces
Use energy methods when: You only need final speed (not time/forces); the problem involves heights, springs, or curved paths; there are too many forces to easily F = ma.
Use F = ma when: You need the acceleration itself; you need time information; you need to find a specific force (like tension or normal force).
Often you need both: Use energy to find a speed, then use F = ma for the acceleration at a particular point.
⚠️ Common exam traps
Don't forget perpendicular forces do ZERO work. Normal force on a sliding block, tension on a swinging pendulum, gravity on a satellite in circular orbit — all do zero work.
KE depends on v², not v. Doubling speed quadruples KE; tripling speed multiplies it by 9.
Don't confuse mass with weight. U_g = mgh uses mass (kg), but the weight (force) of the object is mg. Both appear; track them.
Spring PE depends on x², so direction of stretch/compression doesn't matter. Stretching by +5 cm and compressing by 5 cm give the same U_s.
Energy isn't really "lost" to friction. It becomes thermal energy. Total energy is still conserved if you include heat.
The zero of PE is your choice. Only changes in PE matter. Often the lowest point makes the cleanest problem.
Power ≠ energy. A 100 W bulb on for 1 second uses 100 J. The same bulb on for an hour uses 360,000 J. Same power, different total energy.