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

We will end up reading a single master sentence written in symbols. But we refuse to show it until every ingredient inside it has been defined and drawn. So this page introduces the pieces one at a time — plain words first, a picture next, and only then the symbol. The master sentence appears in full in section 6, and the three concrete PE formulas it produces are written out and derived in section 6b.

1. Position, height, distance — the "where" symbols

The picture: all four are just arrows measuring how far. $h = Δ y$ and $x$ start at a natural "zero" (the ground $y_{0}$ ; the spring at rest). $r$ starts at a planet's core.

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

Figure 1 — Three "where" measures. Left: $h = y - y_{0}$ is the vertical gap (lavender arrow) between the raised object and the floor $y_{0} = 0$ . Middle: $r$ (lavender arrow) runs from the planet's centre out to the object. Right: $x$ (lavender arrow) is the stretch of the spring measured from its relaxed length $x = 0$ . Notice each starts at its own natural zero.

Why the topic needs them: potential energy depends only on position. If we can't name position, we can't write PE. Each formula grabs the position label that fits its geometry: $m g h$ uses $h$ , $- GM m / r$ uses $r$ , $\frac{1}{2} k x^{2}$ uses $x$ .

2. Force and the arrow $F$ — the "push" symbol

The picture: $F = - m g \overset{y}{^}$ reads as "an arrow of length $m g$ pointing in the $- \overset{y}{^}$ direction (downward)." The minus in front just flips the arrow.

3. The dot product $F \cdot d r$ — "how much of the push is along the motion"

The picture below shows why: only the part of the force along the tiny step $d r$ counts.

Figure 2 — Reading a dot product. The coral arrow is the force $F$ (pointing down); the mint arrow is the tiny step $d r$ (pointing up-and-right). The lavender arrow is the piece of $F$ that lies along $d r$ — that projected piece, times the step's length, is $F \cdot d r$ . If $F$ were at a right angle to $d r$ , the lavender arrow would shrink to nothing and the dot product would be zero.

Why this tool and not plain multiplication? Because "work" (defined next) means force helping motion. If you carry a book sideways, gravity (down) is at a right angle to your step (sideways) — it does zero work. Only the dot product captures that "only the aligned part counts" rule. See Work done by a force.

4. Work $W$ , the tiny step $d r$ , and the integral $\int$ — "add up over the whole path"

The picture: slice the path into millions of dots; find the little work $F \cdot d r$ at each dot; pile them all up.

5. The reference point and $Δ U$ — "measured from where?"

Figure 3 — The reference is a free choice. The lavender curve is $U = - GM m / r$ with $U = 0$ placed at infinity; the dashed coral curve is the same physics with the zero moved elsewhere — the whole curve slides up. The two vertical slate arrows show the gap between the same pair of points on each curve: the individual $U$ values differ, but $Δ U$ (the arrow's length) is identical. Only $Δ U$ is physical.

6. The master sentence — and where its minus sign comes from

Now every mark is defined. Read it: $Δ U = - W = - \int_{i}^{f} F \cdot d r$

Only conservative forces (section 8) let us write $U$ at all — that is the licence for the equation above.

6b. Feeding the master sentence — the three concrete $U$ formulas

The master sentence is a machine: put a force in, get a $U$ out. Here we turn its crank three times so you actually see each formula the topic will make you apply.

Why three? Each uses a different position label ( $h$ , $r$ , $x$ ) and a different force shape (constant, inverse-square, linear) — together they cover "flat ground," "deep space," and "stretchy things," which is every case the parent topic asks about. See Conservation of mechanical energy for what these $U$ 's then do.

7. The constants $m$ , $g$ , $k$ , $G$ , $M$ , and the radius $R$ — the "how strong" numbers

Why the topic needs them: they convert "how far" into "how much energy." Bigger $m$ , stiffer $k$ , heavier $M$ → more banked energy for the same move.

8. "Conservative" force — the gatekeeper concept

Why it matters: PE can only be a clean function of position if the banked energy doesn't depend on the route. That is the entire licence to write $U (position)$ . See Conservative vs non-conservative forces.

9. Slope of the landscape — from the integral back to $\frac{d U}{d x}$

Section 6 added up force to get $U$ . Now we run the machine backwards: given the $U$ landscape, recover the force. The reverse of "add up" is "look at the slope."

The picture: draw $U$ as hills and valleys. A ball rolls downhill, toward lower $U$ . The force is minus the slope: steeper hill, harder shove.

Figure 4 — Force is minus the slope of $U$ . The lavender curve is a valley-shaped $U (x)$ . At the coral dot the butter dashed line shows the local slope $d U / d x$ (positive: $U$ rises to the right). The mint arrow is the force $F = - d U / d x$ , pointing the opposite way — downhill, back toward the valley floor. Where the curve is flat (the very bottom) the slope is zero and so is the force.

Intuition The edge case: when position is more than one number

$\frac{d U}{d x}$ only handles motion along one line. In real space you can move in $x$ , $y$ and $z$ , so $U$ depends on all three. A partial derivative $\frac{\partial U}{\partial x}$ (the curly $\partial$ means "derivative in $x$ only, holding $y$ and $z$ frozen") measures the slope along just one axis. Stack the three slopes into one arrow and you get the gradient, written with the upside-down triangle $\nabla$ ("del" or "grad"): $\nabla U = (\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z}) .$ This arrow points the way $U$ increases fastest, and its length is that steepest slope. The full law is then $F = - \nabla U,$ the minus meaning "force points downhill, opposite to steepest increase." Along a single axis only $\partial U / \partial x$ survives and this collapses back to $F_{x} = - \frac{d U}{d x}$ — so the 1-D rule is just the flat-ground version of the real thing, exactly as $m g h$ is for $- GM m / r$ . Built further in Force from potential — F = -dU/dx.

Equipment checklist

Read

F = - m g \overset{y}{^}

aloud in plain words.

"A force of size

m g

pointing straight down."

What does the hat in

\overset{r}{^}

mean?

A unit arrow — length exactly 1 — carrying direction only.

Write

h

in terms of

y

and state which

Δ

it uses.

h = y - y_{0} = Δ y

— the same "final minus initial"

Δ

as in

Δ U

When is

F \cdot d r

zero?

When force and the step are at a right angle (force does no work).

Write work

W

as an integral.

W = \int_{i}^{f} F \cdot d r

— sum of aligned-push × tiny-step over the path.

Why use an integral instead of force × distance?

Because the force changes along the path (weaker far out, stronger when stretched); only slicing-and-adding totals it honestly.

State the master relation between

U

and

W

Δ U = - W

; the minus flips because positive work spends the bank while work against the force fills it.

Write the three concrete PE formulas and their references.

U = m g h

(zero at floor);

U = - GM m / r

(zero at infinity);

U = \frac{1}{2} k x^{2}

(zero at relaxed length).

Why the minus sign in

Δ U = - W

Lifting against gravity gives

W < 0

yet stores energy (

U

up); only

- W

makes those agree.

What does

Δ

Δ U

mean?

"Change in" — final minus initial.

Define

R

and state

g

in terms of

G

M

R

R

= the planet's radius (centre to surface);

g = GM / R^{2}

at the surface.

What makes a force conservative?

Its work is path-independent, so PE can be a single-valued function of position.

Show how

F = - d U / d x

comes from the master integral.

Over a tiny step

d U = - F d x

; divide by

d x

to get

F = - d U / d x

What does

\partial U / \partial x

mean and how does it build

\nabla U

Slope of

U

x

with

y, z

frozen; stacking

\partial U / \partial x, \partial U / \partial y, \partial U / \partial z

gives

\nabla U

, and

F = - \nabla U

Connections

Parent: 1.3.05 Potential energy — definition, gravitational (mgh and −GMm - r), elastic (½kx²) (Hinglish)
Work done by a force
Conservative vs non-conservative forces
Conservation of mechanical energy
Hooke's Law
Gravitation — Newton's law
Escape velocity
Force from potential — F = -dU/dx