Potential energy — definition, gravitational (mgh and −GMm - r), elastic (½kx²)

WHAT is potential energy?

WHY the minus sign? If a force does positive work on an object (helps it move), it spends stored energy, so $U$ goes down. Conversely, if you push against the force, you store energy and $U$ goes up. The minus sign encodes "stored energy = work done against the force."

The inverse relation (force from energy): $F_x = -\frac{dU}{dx} \qquad \vec F = -\nabla U$ WHY: force points "downhill" in the energy landscape, toward lower PE — like a ball rolling into a valley.

HOW to derive each case (from scratch)

1. Gravitational PE near Earth's surface — $mgh$

Derivation. Near the surface the gravitational force is constant: $\vec F = -mg\,\hat y$ (downward). Lift an object from $y=0$ to $y=h$. The force does work: $W_{\text{grav}} = \int_0^h (-mg)\,dy = -mgh$ Then $\Delta U = -W_{\text{grav}} = -(-mgh) = mgh.\;\checkmark$ Why this step? The displacement is upward ($+dy$) while gravity points down, so $\vec F\cdot d\vec r<0$ — gravity does negative work, and we store $+mgh$.

2. General gravitational PE — $-\dfrac{GMm}{r}$

Derivation. Now the force varies: $\vec F = -\dfrac{GMm}{r^2}\hat r$ (attractive, inward). Move a mass from infinity to distance $r$: $W = \int_{\infty}^{r}\left(-\frac{GMm}{r'^2}\right)dr' = -GMm\left[-\frac{1}{r'}\right]_{\infty}^{r} = -GMm\left(-\frac1r + 0\right)=\frac{GMm}{r}$ $U(r) = -W = -\frac{GMm}{r}.\;\checkmark$ Why negative? At infinity $U=0$. As the mass falls inward, gravity does positive work, so $U$ drops below zero. The object is in a "well." You'd need to add energy to climb back out to infinity.

3. Elastic (spring) PE — $\tfrac12 kx^2$

Derivation. Hooke's law: the spring force is $F = -kx$ (restoring, opposes stretch). Stretch slowly from $0$ to $x$: $W_{\text{spring}} = \int_0^x (-kx')\,dx' = -\tfrac12 kx^2$ $U = -W_{\text{spring}} = \tfrac12 kx^2.\;\checkmark$ Why $x^2$ (not $x$)? Because force grows with stretch — stretching the last bit costs more than the first bit. Integrating a linear force gives a quadratic energy. Note $U(+x)=U(-x)$: compressing stores the same as stretching.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Connections

• Work done by a force — PE is just banked work.
• Conservative vs non-conservative forces — only conservative forces define a PE.
• Conservation of mechanical energy — $K+U=\text{const}$.
• Hooke's Law — gives $F=-kx$, integrated into $\tfrac12kx^2$.
• Gravitation — Newton's law — source of $-GMm/r$.
• Escape velocity — uses well depth $GMm/R$.
• Force from potential — F = -dU/dx — the inverse operation.

Concept Map

Hinglish (regional understanding)