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Potential Energy

Source lecture(s): SC133 Lec 9

Intuition

Lift a book: you do work against gravity, and the motion you "paid for" seems gone. But drop the book — the motion comes back, in full. The work wasn't lost; it was banked. Potential energy is stored work: energy of configuration (how high, how stretched, how far apart) rather than of motion. Only special forces offer this banking service — the conservative ones.

Conservative forces & the definition

A force is conservative if the work it does between two points is independent of the path (equivalently: zero work around any closed loop). Gravity and springs qualify; friction does not — a longer path means more heat.

For a conservative force, define potential energy by the work against it:

\[\Delta U = -W_\text{cons} = -\int \vec F\cdot d\vec s\]

and recover the force from the landscape:

\[F_x = -\frac{dU}{dx}\]

— force points downhill on the energy landscape, with magnitude equal to the slope.

The standard catalogue

System Potential energy Reference
Uniform gravity \(U = mgh\) any convenient \(h=0\)
Spring (Hooke) \(U = \tfrac12 kx^2\) natural length
Newtonian gravity \(U = -\dfrac{GMm}{r}\) zero at \(r \to \infty\)

Only differences in \(U\) matter — the zero point is yours to choose (and for \(-GMm/r\) the conventional choice makes bound orbits have \(E < 0\); see gravitation).

Reading energy landscapes

On a \(U(x)\) graph:

  • Equilibria sit where \(dU/dx = 0\) — valleys are stable (restoring force), hilltops unstable.
  • A particle with total energy \(E\) moves where \(E \geq U(x)\); the boundaries are turning points.
  • Near any valley bottom, \(U \approx \tfrac12 k x^2\) — which is why simple harmonic motion is everywhere.

This landscape-reading skill scales all the way up to stability analysis in fluids and plasma equilibria.

Worked example: spring-launched block

A 0.5 kg block compresses a \(k = 800\,\text{N/m}\) spring by 10 cm, then is released on a frictionless track. Launch speed?

\[\tfrac12 kx^2 = \tfrac12 mv^2 \Rightarrow v = x\sqrt{k/m} = 0.1\sqrt{1600} = 4\,\text{m/s}\]

No forces, no trajectory, no time — just the bank transfer \(U \to K\).

Common mistakes

  • Assigning potential energy to friction. Non-conservative forces have no \(U\); their work depends on the path taken.
  • Treating \(U\)'s zero point as physical. Negative potential energy is not weird — only \(\Delta U\) appears in physics.
  • Forgetting whose energy it is. \(U\) belongs to the system (Earth + book, spring
  • block), not to one object alone.
  • Sign slips in \(F = -dU/dx\) — the minus sign is the whole content: force pushes toward lower \(U\).

Knowledge graph position

Prerequisites: Work & kinetic energy. Leads to: Conservation of energy, SHM, orbits.

Quiz

Q1 (computational). A 60 kg hiker climbs 500 m of elevation. Change in gravitational potential energy?

Answer

\(\Delta U = mgh = 60\times9.8\times500 = 2.94\times10^5\,\text{J}\) ≈ 0.3 MJ — about the energy in 70 food-calories. Bodies are inefficient; the hike costs far more.

Q2 (conceptual). Why can't we define a potential energy for friction?

Answer

Friction's work depends on path length, not endpoints — around a closed loop it does nonzero (negative) work. No single-valued function of position can encode that; the energy leaves mechanics as heat.

Q3 (multiple choice). At a stable equilibrium point, \(U(x)\) has: (a) maximum, \(F=0\) (b) minimum, \(F=0\) (c) zero value

Answer

(b). Valley bottom: zero slope (no force) and curvature that pushes back when displaced.