Conservation of Energy
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
When only conservative forces do work, mechanical energy is conserved: K + U = constant. When friction or another non-conservative force acts, its work accounts for the change: Wnc = ΔK + ΔU.
1. The energy equation
2. Combining with forces
Energy gives you speeds, not forces. For “will it stay on the track?” or “what’s the tension?” combine energy (for v) with ΣF = mv²/r.
Speed depends only on height drop, not the ramp’s shape: two frictionless ramps from the same height give the same final speed.
Far from Earth, use U = −GMm/r. To escape, the total energy must reach 0: ½mv² − GMm/R = 0, giving v = √(2GM/R), about 11 km/s for Earth.
Energy isn’t conserved across inelastic collisions; use momentum there.
3. Example, then try it
A spring (k = 800 N/m) compressed 0.15 m launches a 2.0 kg block across a frictionless floor and up a frictionless ramp. Find its launch speed and how high it rises.
½kx² = ½(800)(0.15)² = 9.0 J = ½(2.0)v², so v = 3.0 m/s.
At the top all 9.0 J is mgh: h = 9.0 ÷ 20 = 0.45 m.
A small cart rolls without friction from rest down a track and around a vertical loop of radius R = 0.40 m. What is the lowest starting height (above the bottom of the loop) that keeps it on the track?
Top of the loop: at the minimum speed, gravity alone supplies the centripetal force: mg = mv²/R, so v² = gR.
Energy: mgh = mg(2R) + ½mv² = mg(2R) + ½mgR.
Solve: h = 2R + ½R = 2.5R. Mass and g cancel.
Numbers: h = 2.5(0.40) = 1.0 m.
Answer: h = 2.5R = 1.0 m.
4. Common mistakes
A block slides down a ramp with friction. Which statement about its mechanical energy, K + U, is true?
Friction does negative work, turning mechanical energy into thermal energy, so K + U decreases.
Ready to practice?
5 questions on Conservation of Energy, each with a hint and an explanation for every wrong answer.