PhysicsLesson 11 of 18About 35 minutes

Potential energy

A book on a high shelf is not moving, yet it clearly has the potential to do something: knock it off and it speeds up as it falls. Energy can be stored by position or by shape. This lesson covers the two kinds of stored energy you will use most.

By the end of this lesson you can

explain the difference between kinetic and potential energy;
calculate gravitational potential energy and choose a sensible zero level;
use the spring constant and calculate elastic potential energy;
name the forms of energy in everyday situations.

Stored energy

Think first

An egg rolls off a kitchen counter. Just before it hits the floor it is moving fast, so it has kinetic energy. Where did that energy come from?

Compare your answer

From its height. On the counter, the egg had energy stored because of its position above the floor. As it fell, gravity did work on it and that stored energy turned into kinetic energy.

Definition

Potential energy is energy stored in an object because of its position, shape or condition. It is energy that could be released to make something move.

A drawn bow, a wound-up spring, water behind a dam and a battery all store potential energy. Food stores chemical potential energy, which your body releases to move your muscles. This lesson focuses on two mechanical kinds: gravitational and elastic.

Gravitational potential energy

Lifting an object at constant speed takes work $W = mgh$. That work is stored as gravitational potential energy, and it can be recovered when the object falls back down.

Gravitational potential energy

$$PE_g = mgh$$

$h$ is the height of the object above a zero level that you choose.

Height above what? Above any level you like: the floor, a table top, the ground outside. The value of $PE_g$ depends on the choice, but the change in $PE_g$ as the object moves does not. Physics only ever uses changes in potential energy, so pick whichever zero level makes the numbers easiest, and keep it the same for the whole problem.

Choosing the zero level

Interactive

zero level

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

With the zero at the floor, read $PE_g$ for the ball on the shelf and on the table.
Move the zero to the table, then to the shelf. Which values change? Can $PE_g$ be negative?
Drop the ball to the table with each zero level. Is the change in potential energy the same every time?

Worked example 1

A 0.50 kg ball rests on a shelf 2.0 m above the floor. A table top is 1.0 m above the floor. (a) Find $PE_g$ of the ball with the floor as zero. (b) Find it with the table top as zero. (c) The ball falls from the shelf to the table. Find the change in $PE_g$ using each zero level.

(a) Zero at the floor. $h = 2.0\ \text{m}$: $PE_g = (0.50)(9.81)(2.0) = 9.8\ \text{J}$.
(b) Zero at the table. $h = 1.0\ \text{m}$: $PE_g = (0.50)(9.81)(1.0) = 4.9\ \text{J}$.
(c) The change. Floor as zero: $PE_f - PE_i = 4.9 - 9.8 = -4.9\ \text{J}$. Table as zero: $0 - 4.9 = -4.9\ \text{J}$.
The values of $PE_g$ differ, but the change is $-4.9\ \text{J}$ either way.

Elastic potential energy

Push a block into a spring and let go: the block shoots away. The spring stored energy while it was compressed. Any object that can be stretched or squashed and then spring back, such as a rubber band, a trampoline or a bow, stores elastic potential energy.

The force needed to stretch or compress a spring grows in proportion to how far you move its end from the relaxed position:

$$F = kx$$

Here $x$ is the distance from the relaxed length, and $k$ is the spring constant, in newtons per meter (N/m). A stiff spring has a large $k$; a soft one has a small $k$.

Elastic potential energy

$$PE_{\text{elastic}} = \tfrac12 kx^2$$

$x$ is the stretch or compression measured from the spring's relaxed length, not the spring's total length.

Because $x$ is squared, stretching a spring twice as far stores four times as much energy.

Stretching and compressing a spring

Interactive

$x$

$\vec F_{\text{hand}}$

Stretch $x$Spring constant

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

Stretch the spring by 0.10 m, then 0.20 m. How do the force and the stored energy change?
Look at the graph. The shaded triangle under the force line has area $\tfrac12 (x)(kx)$. What does that area represent?
Compress the spring instead of stretching it. Is the stored energy positive or negative?

Worked example 2

A screen-door spring is 0.20 m long when relaxed, and its spring constant is 60.0 N/m. Opening the door stretches it to 0.35 m. How much elastic potential energy does it store?

Find the stretch, not the length. $x = 0.35\ \text{m} - 0.20\ \text{m} = 0.15\ \text{m}$.
Substitute. $$PE_{\text{elastic}} = \tfrac12 (60.0\ \text{N/m})(0.15\ \text{m})^2$$
$PE_{\text{elastic}} = 0.68\ \text{J}$

Worked example 3

A 2.0 kg bag hangs at rest from a spring with $k = 400\ \text{N/m}$. How far does the spring stretch, and how much elastic energy does it store?

The bag is in equilibrium, so the spring force balances the weight: $kx = mg$.
Stretch. $x = \dfrac{mg}{k} = \dfrac{(2.0)(9.81)}{400} = 0.049\ \text{m}$ (about 4.9 cm).
Energy. $PE_{\text{elastic}} = \tfrac12 (400)(0.049)^2 = 0.48\ \text{J}$.
The spring stretches 4.9 cm and stores 0.48 J.

Check yourself

Question 1

A 1.50 kg dictionary sits on a shelf 2.20 m above the floor. Taking the floor as the zero level, find its gravitational potential energy.

Show solution

$PE_g = (1.50)(9.81)(2.20) = 32.4\ \text{J}$

Question 2

What is the dictionary's potential energy relative to a desk top 0.80 m above the floor?

Show solution

Height above the desk: $2.20 - 0.80 = 1.40\ \text{m}$. $PE_g = (1.50)(9.81)(1.40) = 20.6\ \text{J}$.

Question 3

A 30.0 kg child sits on a swing with ropes 2.50 m long. The swing is pulled back until the ropes make 40.0° with the vertical. What is the child's potential energy relative to the lowest point of the swing?

Show solution

The seat rises by $h = L - L\cos\theta = 2.50(1 - \cos 40.0^\circ) = 0.585\ \text{m}$.

$PE_g = (30.0)(9.81)(0.585) = 172\ \text{J}$

Question 4

A pogo stick's spring has $k = 5.0 \times 10^3\ \text{N/m}$. A child compresses it by 12 cm. How much elastic energy is stored?

Show solution

$x = 0.12\ \text{m}$: $PE = \tfrac12 (5.0 \times 10^3)(0.12)^2 = 36\ \text{J}$

Question 5

You compress a spring by 2 cm and it stores 0.10 J. How much does it store if you compress it by 6 cm?

0.30 J
0.60 J
0.90 J
1.80 J

Show solution

Three times the compression gives $3^2 = 9$ times the energy: $9 \times 0.10 = 0.90\ \text{J}$.

Question 6

Name the main forms of energy: (a) a cyclist coasting on a level road; (b) an archer holding a fully drawn bow; (c) a rock balanced at the top of a cliff; (d) a pan of water warming on a stove.

Show solution

(a) Kinetic. (b) Elastic potential, in the bent bow. (c) Gravitational potential. (d) Thermal energy, increasing as heat flows in.

Summary

Potential energy is energy held in storage by where an object is, how it is shaped, or its condition.
Gravitational: $PE_g = mgh$. Pick any zero level for $h$; only the changes in $PE_g$ carry meaning.
Springs: $F = kx$ and $PE_{\text{elastic}} = \tfrac12 kx^2$, with $x$ measured from the relaxed length.

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