3.3 Potential energy

Four little picture stories

Press Next (or Play) to walk through each story one small step at a time. The numbers come last.

1. Lifting a backpack onto a shelf

2. A stretched spring stores energy

You lift the same backpack onto a shelf three times as high. The stored energy becomes:

3. Who is right about the ball on the roof?

Read the story as text

Maya stands on the roof of a 5.0 m shed holding a 0.50 kg ball. Her friend on the ground says: "That ball has 24.5 J of potential energy." Maya says: "No, it is sitting right here at my level. It has zero."

Then Maya drops it. Both of them agree on how fast it hits the ground. So what is potential energy, and how can two people give it different values and still get the same physics?

Two friends pick different zero levels for a falling ball. Which do they always agree on?

4. Check yourself

Think of your answer first, then tap to see it.

a) You lift the same backpack onto a shelf twice as high. How much more energy is stored?

Show answer

Twice as much. Height counts once: Ug = m g h.

b) You stretch a spring three times as far. How much more energy does it store?

Show answer

9 times as much. Stretch counts squared: 3 × 3 = 9.

c) A 2.0 kg backpack is lifted 1.5 m onto a shelf. How much does Ug of the backpack-Earth system go up? (g = 9.8 m/s²)

Show answer

ΔUg = m g h = 2.0 × 9.8 × 1.5 = 29.4 J.

Already know this?

Three quick questions. Get all three right on the first try and you can skip ahead.

A 2.0 kg box is lifted 3.0 m. By how much does Ug of the box-Earth system go up? (g = 9.8 m/s²)

A spring is stretched twice as far. Its stored energy becomes:

Gravitational potential energy belongs to:

Energy stored in the arrangement of a system: gravity near Earth (m g h), springs (½ k x²), where to put "zero", and why potential energy belongs to a system, not to one object.

The idea

In pictures

  • Lift a backpack onto a shelf: energy is stored in the backpack + Earth. Let it fall and the energy comes back as motion.
  • Higher shelf or heavier bag: more stored energy. Twice as high, twice the energy.
  • Stretch or squash a spring: energy is stored in the spring. Twice the stretch, four times the energy.
  • You may choose where "zero height" is. The stored numbers change, but the change during a fall never does.

Below is the same idea in words, and then with numbers.

Potential energy (U) is energy stored in how the parts of a system are arranged: how far a ball is from Earth, how much a spring is squashed. When the arrangement changes, stored energy turns into other kinds (like kinetic energy) or back.

It belongs to a system

Gravitational potential energy is energy of the ball + Earth system. It comes from the pull between them. A ball on its own cannot have potential energy. This choice changes how you do the bookkeeping:

Both give the same answer. Never use both Ug and Wg at once, or you count gravity twice.

Gravity near Earth's surface

ΔUg = m g Δy

Raise an object and the system's Ug goes up; lower it and Ug goes down. Only changes matter, so you may pick any height as y = 0. Then Ug = m g y. Below your zero, Ug is negative, and that is fine.

Springs

Us = ½ k x²

x is how far the spring is stretched or compressed from its natural (relaxed) length, not the spring's length. k is the spring constant in N/m. Us is never negative: stretched or squashed, it stores energy. Double x and Us goes up 4 times. This ½ k x² is exactly the work you did to stretch it (the triangle area in 3.2).

Far from Earth (the general form)

For two objects far apart (a planet and a satellite), g is not constant. The equation sheet gives

UG = − G m1 m2 / r

Here zero is chosen at r = infinity (very far apart), so UG is always negative and gets less negative (goes up) as the objects move apart. Near Earth's surface its changes match m g Δy.

Conservative forces. Gravity and springs are "conservative": the work they do depends only on the start and end arrangement, not on the path. That is why we can store their energy as U, with Winternal = −ΔU. Friction is not conservative: a longer path means more friction work, so there is no "friction potential energy".

With numbers: Maya's ball and a spring

Show all steps as text
  1. Friend's zero at the ground: Ug = m g y = 0.50 × 9.8 × 5.0 = 24.5 J on the roof; 0 J at the ground. y measured from the ground.
  2. Maya's zero at the roof: Ug = 0 J on the roof; at the ground y = −5.0 m, so Ug = 0.50 × 9.8 × (−5.0) = −24.5 J. Below the zero line, y is negative.
  3. Change during the fall: friend 0 − 24.5 = −24.5 J; Maya −24.5 − 0 = −24.5 J. The same. Only ΔU is physical, so both predict the same speed.
  4. Speed at the ground: the 24.5 J becomes K: ½ (0.50) v² = 24.5, v² = 98, v = 9.9 m/s. Preview of 3.4: lost U becomes K.
  5. A spring with k = 400 N/m compressed 0.10 m: Us = ½ × 400 × (0.10)² = 2.0 J. Compressed 0.20 m: ½ × 400 × 0.04 = 8.0 J. Twice the compression, four times the energy.

Play: choose your zero, and a spring

Drop a ball. Move the "zero level" slider and watch the Ug bar: its value changes, can even go negative, but the change during the fall, and the final speed, never do.

A block on a frictionless floor bounces on a spring. The spring's stored energy and the block's kinetic energy trade back and forth.

Worked examples

A hiker climbs basic

A 60 kg hiker climbs 300 m up a hill by a winding trail. By how much does the hiker-Earth gravitational potential energy change?

Show solution
Show all steps as text
  1. ΔUg = m g Δy = 60 × 9.8 × 300 = 176 400 J ≈ 1.8 × 10⁵ J. Only the height gain counts.
  2. The winding trail does not matter. Gravity is conservative: path does not matter.

A stretched spring basic

A spring with k = 250 N/m is stretched 0.20 m. How much energy does it store?

Show solution
Show all steps as text
  1. Us = ½ k x² = ½ × 250 × (0.20)² = ½ × 250 × 0.04 = 5.0 J. Square x first.

Two zero levels medium

A 2.0 kg book sits on a shelf 1.5 m above the floor. A table top is 0.80 m above the floor. Find Ug of the book-Earth system with zero at the floor and with zero at the table top. Then find ΔUg when the book is moved from the shelf to the table top.

Show solution
Show all steps as text
  1. Floor zero: U = 2.0 × 9.8 × 1.5 = 29.4 J. y = 1.5 m.
  2. Table zero: y = 1.5 − 0.80 = 0.70 m, U = 2.0 × 9.8 × 0.70 = 13.7 J. Measure y from the chosen zero.
  3. Shelf to table: floor zero 2.0 × 9.8 × 0.80 − 29.4 = 15.7 − 29.4 = −13.7 J; table zero 0 − 13.7 = −13.7 J. The change is the same either way.

Finding k from stored energy medium

A spring stores 3.0 J when compressed 0.10 m. Find k. How much does it store when compressed 0.30 m?

Show solution
Show all steps as text
  1. k = 2 U / x² = 2 × 3.0 / (0.10)² = 6.0 / 0.010 = 600 N/m. Rearranged U = ½kx².
  2. At 0.30 m (3 times the compression): U = 3² × 3.0 = 27 J. Check: ½ × 600 × 0.09 = 27 J. U ∝ x².

Same fall, two systems AP

A 0.50 kg ball falls 2.0 m from rest (no air resistance). Describe the energy of (a) the ball + Earth system and (b) the ball alone as a system, and find the ball's final kinetic energy each way.

Show solution
Show all steps as text
  1. (a) Ball + Earth: no outside forces do work, so total energy stays constant. Ug falls by m g h = 0.50 × 9.8 × 2.0 = 9.8 J, so K rises by 9.8 J. Gravity is internal; its effect is the drop in Ug.
  2. (b) Ball alone: no Ug exists. Earth's gravity is an outside force that does W = m g h = +9.8 J on the ball, so K rises by 9.8 J. Work-energy theorem.
  3. Both: K = 9.8 J at the bottom (v = √(2 × 9.8 / 0.50) = 6.3 m/s). Different bookkeeping, same physics.

Practice

  1. A spring is compressed 3 times as far as before. Its stored energy becomes

    Worked answer

    Us = ½kx², so 3 times x gives 3² = 9 times Us.

  2. Which of these can have gravitational potential energy?

    Worked answer

    Potential energy belongs to a system of objects that interact. Gravitational U needs both the ball and Earth.

  3. A 1.0 kg ball rests on the floor. The zero of Ug is chosen at a table top 0.80 m above the floor. What is Ug of the ball-Earth system?

    Worked answer

    y = −0.80 m relative to the chosen zero: U = 1.0 × 9.8 × (−0.80) = −7.84 J ≈ −7.8 J.

  4. Which graph shows spring potential energy Us (vertical) against displacement x from the natural length, for x from −0.2 m to +0.2 m?

    Worked answer

    U = ½kx² is the same for +x and −x and is zero at x = 0: a U-shaped parabola.

  5. A 4.0 kg box is lifted from a shelf 1.0 m high to a shelf 3.0 m high. ΔUg of the box-Earth system is

    Worked answer

    Δy = 3.0 − 1.0 = 2.0 m. ΔU = 4.0 × 9.8 × 2.0 = 78.4 J ≈ 78 J (positive: it went up).

  6. A satellite moves from an orbit of radius r to a larger radius 2r around a planet. What happens to UG = −G M m / r of the planet-satellite system?

    Worked answer

    At 2r, UG = −GMm/(2r), which is half as negative as −GMm/r. A less negative number is bigger, so UG increased, just as lifting something raises Ug.

  7. Short answer. Explain why two students who choose different zero levels for gravitational potential energy still predict the same speed for a falling rock.

    Worked answer

    Changing the zero adds the same constant to every Ug value. Speeds come from the change in Ug (ΔU = m g Δy), and the constant cancels when you subtract. So ΔU, and the K gained, are the same for both.

  8. Short answer (linearize). A student measures the energy stored in a spring: x = 0.10 m, U = 0.50 J; x = 0.20 m, U = 2.0 J; x = 0.30 m, U = 4.5 J. What should she plot to get a straight line? Find k.

    Worked answer

    Plot U against x². x² = 0.010, 0.040, 0.090 m² and U = 0.50, 2.0, 4.5 J: a straight line through the origin with slope 0.50 / 0.010 = 50 J/m².

    Slope = ½ k, so k = 100 N/m. Check: ½ × 100 × 0.30² = 4.5 J.

AP question types for this topic

These four free response questions match the four question types on the AP Physics 1 exam, each sized for potential energy. Work each part on paper first, then open the worked answer to check it and see how points are given.

1. Mathematical Routines

Derive with symbols first, then put in numbers. Zara hangs a block of mass m from a vertical spring. At rest, the spring is stretched by x0.

  1. Derive expressions for the spring constant k and the elastic potential energy stored, in terms of m, g and x0.
  2. With m = 0.50 kg and x0 = 0.080 m, calculate k and the stored energy.
  3. She hangs a second identical block below the first. Derive by what factor the stored spring energy changes, and give the new value.
Worked answer and scoring

(a) At rest the spring force balances the weight: kx0 = mg, so k = mg/x0. Us = ½kx0² = ½mgx0.

(b) k = 0.50 × 9.8 / 0.080 = 61 N/m. Us = ½ × 0.50 × 9.8 × 0.080 = 0.20 J.

(c) Twice the weight gives twice the stretch, 2x0. Us = ½k(2x0)² = 4 × ½kx0²: 4 times as much, 0.78 J.

Scoring (6 points)

  • 1 point for kx0 = mg
  • 1 point for Us = ½mgx0
  • 1 point for 61 N/m
  • 1 point for 0.20 J
  • 1 point for the stretch doubling
  • 1 point for 4 times, 0.78 J

2. Translation Between Representations

Move between graphs, pictures, equations and words, and justify. Ravi studies two systems: a spring stretched by x, and a ball-Earth system with the ball at height y near the ground.

Four graph shapes; the vertical axis is potential energy U:

UA
Graph A: a straight line up through the origin.
UB
Graph B: starts flat at the origin and curves up more and more steeply.
UC
Graph C: a flat horizontal line.
UD
Graph D: a straight line going down.
  1. Which graph shows the spring's U against stretch x? Justify with an equation.
  2. Which graph shows the ball-Earth U against height y (zero at the ground)? Justify.
  3. The spring graph passes through (0.10 m, 0.25 J) and the gravity graph has slope 4.9 J/m. Find k and the ball's mass.
Worked answer and scoring

(a) Graph B. Us = ½kx² grows with x², a parabola that gets steeper.

(b) Graph A. Ug = mgy is proportional to y: a straight line through the origin with slope mg.

(c) k = 2U/x² = 2 × 0.25 / 0.10² = 50 N/m. Slope = mg, so m = 4.9 / 9.8 = 0.50 kg.

Scoring (5 points)

  • 1 point for B with U = ½kx²
  • 1 point for A with U = mgy
  • 1 point for slope = mg
  • 1 point for 50 N/m
  • 1 point for 0.50 kg

3. Experimental Design and Analysis

Plan a measurement, make a straight-line graph and read its slope. Lina wants to find how much energy a spring stores when stretched 0.20 m. She has the spring, a force sensor, a metre stick and a clamp stand.

Her data:

stretch x (m)0.050.100.150.20
force F (N)1.94.15.98.0
  1. Describe her procedure.
  2. What should she plot, what does the slope mean, and how does the graph give the stored energy?
  3. Find k and the energy stored at 0.20 m.
Worked answer and scoring

(a) Clamp one end of the spring and note its relaxed length on the metre stick. Pull the other end with the force sensor, hold still at several stretches, and record F and x. Repeat each stretch and average.

(b) Plot F against x: F = kx is a straight line with slope k. The work to stretch the spring is the area under F-x, and that work is the stored energy: area = ½kx².

(c) Best-fit slope ≈ 8.0/0.20 = 40 N/m. U = ½ × 40 × 0.20² = 0.80 J, the same as the triangle area ½ × 0.20 m × 8.0 N.

Scoring (6 points)

  • 1 point for measuring F with the sensor
  • 1 point for measuring x from the relaxed length, with repeats
  • 1 point for plotting F against x
  • 1 point for slope = k ≈ 40 N/m
  • 1 point for area = stored energy
  • 1 point for 0.80 J

4. Qualitative/Quantitative Translation

Explain a claim in words, back it with a derivation, and connect the two. A 0.40 kg ball falls 6.0 m from a roof to the ground. Felix says: Potential energy depends on where you put zero, so the energy the ball turns into kinetic energy also depends on where you put zero.

  1. In words, say whether Felix is right.
  2. Derive ΔU with zero at the ground and with zero at the roof, and calculate both.
  3. Connect (b) to (a).
Worked answer and scoring

(a) No. The value of U depends on the zero, but only the change in U turns into kinetic energy, and the change is the same whatever zero you pick.

(b) ΔU = mg(yf − yi). Zero at ground: 0 − 0.40 × 9.8 × 6.0 = −23.5 J. Zero at roof: yi = 0, yf = −6.0 m, ΔU = 0.40 × 9.8 × (−6.0) = −23.5 J.

(c) Shifting the zero adds the same constant to yf and yi, and it cancels in the difference. Both choices give 23.5 J of kinetic energy, as argued in (a).

Scoring (5 points)

  • 1 point for “no”
  • 1 point for: only changes in U matter
  • 1 point for ΔU = mg(yf − yi)
  • 1 point for −23.5 J both ways
  • 1 point for the constant cancelling

Common mistakes

The mistake: "The ball has potential energy."

Why it is wrong: potential energy comes from an interaction, so it belongs to the ball + Earth system.

How to spot it: on AP questions, check what the system is. If Earth is not in the system, use work by gravity instead.

The mistake: counting both Ug and the work done by gravity.

Why it is wrong: they are two ways to describe the same effect, so you count it twice.

How to spot it: your answer has double the expected energy. Pick one view.

The mistake: using the spring's total length for x, or making Us negative for a compressed spring.

Why it is wrong: x is the change from the natural length; x² is positive either way.

How to spot it: a relaxed spring must give Us = 0.

The mistake: using the distance along a ramp as h in m g h.

Why it is wrong: Ug depends on vertical height only. A 3 m ramp at 30° rises only 1.5 m.

How to spot it: on a slope, use h = d sin θ.