In a Conservative Field Only the End Points Matter
A conservative force does work that depends only on the start and end points, and so has a potential energy. That one property gives path independence, zero work around a closed loop, forces from potentials, and the spring energy accounts below.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Path independence and force from potential
In a conservative field the work done is the same along every path between two points and is zero around a closed loop. If P to Q needs 8 J and Q to R needs 2 J, then P to R needs 10 J. A constant force such as (3i + 4j) N is conservative, so moving from (0, 0) to (a, a) along any route gives 7a joules. The force follows from the potential energy as F = -∇U = -(∂U/∂x, ∂U/∂y, ∂U/∂z): for U = 20xy/z, F = -(20y/z) i - (20x/z) j + (20xy/z²) k, where the last term is positive because the derivative of 1/z is negative.
Springs and energy
The energy stored in a spring is (1/2) k x².
For a ball falling a height h onto a spring that then compresses by d, gravity does m g (h + d) and the spring does -(1/2) k d², so the net work is m g (h + d) - (1/2) k d², and at the lowest point the net work is zero because the kinetic energy is zero. A block released from rest on a vertical spring that starts unstretched falls until m g x = (1/2) k x², so the maximum extension is 2 M g / k, twice the equilibrium extension M g / k.
OlympIQ exam lens
- ✓Conservative force: work independent of path; zero for a closed loop
- ✓F = -∇U; differentiate each coordinate separately and watch the signs
- ✓Spring energy (1/2) k x²; use energy conservation for maximum compression
- ✓Dropped onto a spring: total fall is h + d; at the lowest point K = 0
- ✓Block released on an unstretched hanging spring: maximum extension is 2Mg/k
Do not confuse the equilibrium extension with the maximum extension of a block released from an unstretched spring. The block overshoots to twice the equilibrium value, 2Mg/k, because energy is conserved and it has no friction to stop it halfway.
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