Physics I: Mechanics › Work & energy › full formula sheet

Ug = mgh

Say it: “gravitational potential energy equals mass times g times height”

Gravitational potential energy

The energy of being high up — a bookkeeping trick that turns gravity's work into stored energy you can spend later.

m is mass (kg), g = 9.8 m/s² near Earth's surface, h the height above your chosen reference level (m). Ug is in joules — and only changes in h matter.

Before this lesson: Work (constant force)

Where it comes from

Lift a book to a shelf and you've done work against gravity. Set it down gently and that work seems gone — until the book slides off and lands with exactly the energy you put in. The work wasn't destroyed; it was stored.

Before reading on: you lift 2 kg straight up 3 m, or carry it up a long winding staircase to the same 3 m height. Which trip stores more gravitational energy? Think before you read.

Same energy. Gravity only cares about the vertical change: on the staircase, the extra horizontal walking is perpendicular to gravity (zero work), and the steeper-vs-shallower trade of force-vs-distance cancels exactly. Path doesn't matter — only Δh.

The bookkeeping: gravity does work Wg = −mgΔh on the way up (force down, displacement up). Define Ug so that this work is stored rather than lost:

Ug = mghlifting stores mgh; falling spends it back as kinetic energySay it: “U sub g equals m g h”

Derivation

Gravity is constant (mg downward), so its work is the constant-force rule. Move a mass from h₁ to h₂:

Wg
=
(mg)(h₁ − h₂)
Step 1 — gravity's work. Force mg downward; displacement is (h₂ − h₁) upward. Down × up = negative: Wg = mg × (−(h₂ − h₁)).
=
−(mgh₂ − mgh₁)
Step 2 — regroup. The work is minus the change of the quantity mgh. So define Ug = mgh and write Wg = −ΔUg.
Wg
=
−ΔUg
Step 3 — the payoff. When gravity does positive work (falling), Ug decreases — potential converts to kinetic. When you lift, you do positive work and Ug increases. Energy is conserved in the swap. ∎

The reference level is free. h = 0 can be the floor, the table, or sea level — Ug values shift, but ΔUg (the only thing physics uses) never changes. Pick whatever makes h easy.

How to use it

The procedure, every time:

  1. Choose h = 0. Floor, tabletop, launch point — wherever heights are simplest. Write it down.
  2. Measure h from there. Heights above give positive Ug; below give negative Ug (fine — only differences matter).
  3. Compute Ug = mgh. g = 9.8 m/s² unless told otherwise.
  4. For processes, use ΔUg = mgΔh. Falling (Δh < 0) loses potential; rising gains it.
  5. Remember the sign rule: Wg = −ΔUg. Gravity's work and potential change are opposites.
Common mistake: using the distance traveled (e.g. the staircase length) instead of the vertical Δh. Gravity is vertical — only vertical displacement changes Ug.

Worked examples

Four problems, easiest first. Watch the reference level and the signs.

Example 1 — basic: m = 2 kg on a 5 m shelf (floor is h = 0)

  1. Identify h. h = 5 m above the floor.
  2. Compute. Ug = 2 × 9.8 × 5 = 98 J.
  3. Interpret. Lifting it stored 98 J; dropping it converts all 98 J to kinetic (ignoring air).
Common mistake: answering 98 J but with h measured from the wrong place — e.g. the shelf is 5 m above the floor but you set h = 0 at the table (4 m up), giving U = 19.6 J. State your reference first.
Your turn — m = 3 kg, h = 4 m. Ug = ?

Answer: 117.6 J. 3 × 9.8 × 4 = 117.6 J.

Example 2 — path independence: 5 kg lifted 2 m straight up vs up a ramp

  1. Straight up. ΔUg = 5 × 9.8 × 2 = 98 J.
  2. Up the ramp. Same vertical rise: Δh = 2 m, so ΔUg = 98 J — identical.
  3. Why. On the ramp, gravity's component along the slope is smaller but the distance is longer; force × distance cancels to mgΔh. Only the vertical change survives.
Common mistake: computing W = mg × (ramp length). The ramp is longer than 2 m, so this overcharges. Gravity only bills for vertical meters.
Your turn — 4 kg raised 3 m vertically. ΔUg = ? (Any path.)

Answer: 117.6 J. 4 × 9.8 × 3 = 117.6 J, path-independent.

Example 3 — below the reference: m = 5 kg, h = −2 m (a pit, floor is h = 0)

  1. Compute. Ug = 5 × 9.8 × (−2) = −98 J.
  2. Interpret. Negative potential is fine — it just means “98 J below the reference.” Falling from the floor into the pit gains 98 J of kinetic.
  3. Check differences. From h = 3 m (U = 147 J) to h = −2 m: ΔUg = −245 J — the reference cancels out of the difference.
Common mistake: panicking at negative energy and flipping the sign. Negative Ug is bookkeeping, not an error — only changes are physical.
Your turn — m = 4 kg at h = −3 m. Ug = ?

Answer: −117.6 J. 4 × 9.8 × (−3) = −117.6 J.

Example 4 — energy released: 10 kg falls from h = 20 m to h = 5 m

  1. Change in height. Δh = 5 − 20 = −15 m.
  2. Change in potential. ΔUg = 10 × 9.8 × (−15) = −1470 J.
  3. Where it went. Wg = −ΔUg = +1470 J — gravity did 1470 J of positive work, becoming kinetic energy (the work–energy theorem in action).
Common mistake: computing U at 20 m (1960 J) and stopping — answering the stored energy instead of the released energy. “How much is released” asks for the change.
Your turn — 2 kg drops from h = 10 m to h = 4 m. ΔUg = ?

Answer: −117.6 J. 2 × 9.8 × (−6) = −117.6 J released.

Memorization tips

  • Say it aloud: “U sub g equals m g h — only the vertical change counts.”
  • h = 0 is yours to choose. Pick the level that makes the arithmetic easiest; differences don't care.
  • Vertical only: stairs, ramps, winding paths — gravity bills vertical meters alone. Horizontal distance is free.
  • The sign duet: Wg = −ΔUg. Falling: gravity works positive, potential drops. Lifting: you work positive, potential rises.
  • Negative U is fine. Below the reference is negative by definition — it's a basement, not a bug.
  • Units check: kg · (m/s²) · m = N·m = J. If it isn't joules, recheck.

Final challenge

Five mixed questions — reference levels, paths, and signs. Score 5/5 and gravitational potential is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

Does the choice of h = 0 change the physics?

No. Shifting the reference adds the same constant to every Ug, and every physical prediction uses differences ΔUg, where the constant cancels. Choose h = 0 for convenience.

Why doesn't the path matter?

Gravity is vertical and constant, so its work over any path is mg times the vertical displacement — horizontal detours contribute zero (perpendicular) and slope trade-offs cancel. Mathematically, gravity is a conservative force.

Can gravitational potential energy be negative?

Yes, whenever h is below your reference level. It's just bookkeeping — 'below zero' means 'below the arbitrary line you drew.' Only changes ΔUg carry physical meaning.

Is U = mgh exact?

Only near Earth's surface, where g ≈ 9.8 m/s² is effectively constant. For satellites or interplanetary distances you need U = −GMm/r, where g varies with r.

How does this connect to kinetic energy?

A falling object trades Ug for KE: ΔKE = −ΔUg (gravity's work becomes motion). That's the work-energy theorem with gravity's work rewritten as a potential — the seed of conservation of mechanical energy.

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