Conservation Conditions
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
Three different quantities, three different conditions. Linear momentum: no net external force. Angular momentum: no net external torque. Mechanical energy: no net external work and no conversion to internal energy. Check each one separately.
1. Force vs. torque
2. Check each situation
3. Example, then try it
Why is the ball rolling down a ramp different from a box sliding down a rough ramp?
The rolling ball’s contact point isn’t sliding, so static friction does no work: mechanical energy ✓.
The sliding box rubs along the ramp, so kinetic friction turns energy into heat: mechanical energy ✗.
A merry-go-round (I = 500 kg·m², radius 2.0 m) is at rest on a fixed axle. A 50 kg student runs at 4.0 m/s along the edge’s tangent and jumps on at the rim. Find the final spin rate. Which quantities were conserved?
Angular momentum ✓ (the axle force makes no torque): student’s L = mvr = (50)(4.0)(2.0) = 400 kg·m²/s.
After: I = 500 + (50)(2.0)² = 700 kg·m², so ω = 400 ÷ 700 ≈ 0.57 rad/s.
Linear momentum ✗: the axle pushes on the merry-go-round, so the system’s linear momentum changes.
Kinetic energy ✗: before, ½(50)(4.0)² = 400 J; after, ½(700)(0.57)² ≈ 114 J.
Answer: about 0.57 rad/s. Only angular momentum was conserved.
4. Common mistakes
A diver leaves the board spinning and pulls into a tuck. Ignoring air resistance, which quantity is conserved about her center of mass while she’s in the air?
Gravity acts at her center of mass, so it makes no torque about it: L is conserved, and tucking speeds up her spin. Gravity is an external force, so her linear momentum changes, and her rotational kinetic energy increases as she tucks.
Ready to practice?
4 questions on Conservation Conditions, each with a hint and an explanation for every wrong answer.