Nonconservative Forces & Energy Accounting
- Distinguish conservative from nonconservative forces by the path-independence test
- Apply the generalized work–energy theorem including dissipated energy
- Track energy through a multi-stage problem without losing a term
The test is path independence
A force is conservative if the work it does between two points is the same along every path — equivalently, if the work around any closed loop is zero. Gravity and spring forces pass this test, which is exactly why a potential energy function can be defined for them: U depends only on position. Friction and drag fail it: dragging a block around a loop back to its start does negative work the whole way, so no potential energy function exists for friction, and its effect must be tracked separately.
Energy is not lost, it is relocated
Saying friction "destroys" energy is loose and leads to errors. The energy is converted to thermal energy in the surfaces and eventually radiated away — the total is conserved, but it has left the mechanical account and cannot be recovered as organized motion. This is why the useful formulation is a balance sheet rather than a conservation statement: write what the system had, add what was put in, subtract what was dissipated, and the remainder is what it has.
Path length, not displacement
Because friction always opposes motion, the energy it dissipates is f_k times the total distance traveled along the surface. A block that slides 3 m forward and 3 m back dissipates f_k(6 m) even though its displacement is zero. This is the single most common error in multi-stage energy problems, and it is easy to catch: any time the object reverses direction, add the path segments rather than subtracting them.
A 2.0 kg block is launched at 6.0 m/s along a horizontal surface with μ_k = 0.25, and slides to rest. Find the distance it travels and confirm the energy balance.
- 1.Initial kinetic energy: K = ½(2.0)(6.0)² = ½(2.0)(36) = 36 J.
- 2.Friction force: f_k = μ_k mg = 0.25(2.0)(9.8) = 4.9 N.
- 3.All kinetic energy is dissipated: 36 J = f_k d = 4.9d, so d = 36/4.9 ≈ 7.3 m.
- 4.Check with kinematics: a = −μ_k g = −2.45 m/s², and v² = v₀² + 2ad gives 0 = 36 − 4.9d, the same equation.
A block slides down a rough incline, then back up to its starting height is impossible. The best explanation is that —
Free-response energy questions are graded on the accounting, not the arithmetic. Write an explicit initial-equals-final statement naming every term before substituting numbers. A correct number arrived at without a stated energy equation routinely loses the setup points; a stated equation with a small numerical slip usually keeps most of them.
The energy dissipated by kinetic friction is calculated using —
Answer the 2 checkpoints as you read.
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