C3.1 Relate Hooke’s law, work, and elastic potential energy

Ontario Grade 12 Physics

Energy and Momentum

How a spring’s force and energy change as it stretches or compresses

In SPH3U, you studied force, work, and energy. Force is a vector: it has magnitude and direction. Work and energy are scalars: they have magnitude but no direction. This lesson connects those ideas for an ideal spring. The physical system is a spring and, when useful, the object attached to it. Use a reference frame fixed to the spring’s support. Set the spring’s relaxed position as

x=0

and choose right as the positive direction. A stretch to the right has positive displacement; a compression to the left has negative displacement. The spring’s force points back toward the relaxed position.

What you will learn

Describe the relationship between a spring’s displacement and its force within its elastic range.
Calculate the work done by a spring or by an external force as a spring changes length.
Relate work to changes in elastic potential energy, using correct signs and SI units.

1. Hooke’s law: force depends on displacement

A spring’s relaxed length is its length when it is neither stretched nor compressed. Displacement

x

is the change in the spring’s length from that relaxed position, measured along the spring. It is a signed quantity, so its sign tells which way the spring is displaced.

Within its elastic range, an ideal spring follows Hooke’s law. The spring force is proportional to displacement, but points in the opposite direction. The constant

k

is the spring constant. It measures stiffness: a larger

k

means more force is needed for the same displacement. Its SI unit is newtons per metre,

\mathrm{N/m}

.

The minus sign matters. If the spring is stretched to the right, its force points left. If compressed to the left, its force points right. This force is a restoring force because it acts toward the relaxed position. The law is a model for the spring’s elastic range; do not assume it applies if the spring is permanently deformed.

The equation gives the force exerted by the spring. If an external hand slowly stretches the spring, the hand’s force points in the direction of the stretch and has the opposite sign to the spring force.

F_{s,x}=-kx

Displacement is measured from the relaxed position.
Spring force and displacement have opposite directions.
Use metres for displacement, newtons for force, and newtons per metre for $k$ .

2. Work and the spring’s changing force

Work describes energy transferred when a force acts through a displacement. For a constant force along a straight path, work is force multiplied by displacement in the force’s direction. Work is measured in joules, where

1\,\mathrm{J}=1\,\mathrm{N\,m}

.

A spring’s force is not constant over a stretch from the relaxed position: its magnitude increases as the displacement increases. For a Hooke’s-law spring, the force changes evenly from its initial value to its final value. Therefore, the work can be found using the average of the initial and final spring forces multiplied by the displacement. This algebraic method avoids treating the force as constant.

Work done by the spring is positive when the spring force and the object’s displacement point in the same direction. It is negative when they point in opposite directions. For example, while a stretched spring is released toward equilibrium, the spring force and motion point the same way, so the spring does positive work. While an external agent stretches it slowly, the spring force opposes the displacement, so the spring does negative work.

Elastic potential energy is energy stored in a deformed spring. Taking the relaxed position as zero elastic potential energy, the stored energy depends on the square of displacement. It is non-negative for either a stretch or a compression. The work done by the spring equals the negative change in this stored energy. If an external agent slowly deforms the spring without changing its kinetic energy, the agent’s work equals the increase in elastic potential energy.

W_s=-\Delta E_{\mathrm{el}}=\frac{1}{2}k(x_i^2-x_f^2),\qquad E_{\mathrm{el}}=\frac{1}{2}kx^2

Work and elastic potential energy are scalars measured in joules.
The spring does negative work when its displacement is against the spring force.
A stretch and a compression of equal size store equal elastic potential energy.

3. Choosing the right work relationship

The subscripts

i

and

f

mean initial and final. To find work done by the spring between two positions, use the initial and final displacements in the spring-work relationship. A negative result means energy has been stored in the spring; a positive result means stored energy has decreased.

To find work done by an external force during a slow stretch or compression, compare the initial and final elastic potential energies. If the spring starts relaxed and ends deformed, the external work is positive. If the spring returns to its relaxed position under an external force that controls its motion, that force does negative work.

State the physical system and sign convention before calculating. Keep displacement signs for force and direction questions. For energy, use the square of displacement, which removes its sign. Check that a calculated work or energy is in joules and that its sign matches the described transfer.

W_{\mathrm{ext}}=\Delta E_{\mathrm{el}}=\frac{1}{2}k(x_f^2-x_i^2)

Use signed displacement in Hooke’s law.
Use squared displacement for elastic potential energy.
Clearly name whether the work is done by the spring or by an external agent.

Worked example

Finding the spring force

A spring has

k=180\,\mathrm{N/m}

. Its relaxed position is the origin, with right positive. It is stretched

0.075\,\mathrm{m}

to the right. Find the spring force.

Set the system and direction
The system is the spring and the attached object. The reference frame is fixed to the support. Right is positive, so the stretch is $x=+0.075\,\mathrm{m}$ . The unknown is the spring force along the spring.
Apply Hooke’s law
The spring force is opposite to displacement. Substitute the stiffness and signed displacement.
$F_{s,x}=-(180\,\mathrm{N/m})(+0.075\,\mathrm{m})$
Report and check
The result is negative, so the force points left. The units reduce to newtons. A spring stretched to the right should pull back toward its relaxed position, so the direction is reasonable.
$F_{s,x}=-13.5\,\mathrm{N}$

Answer: The spring exerts a force of

13.5\,\mathrm{N}

to the left.

Check: The magnitude is proportional to the stretch, and the negative sign agrees with the chosen right-positive direction.

Worked example

Work done by a spring while it returns

A spring with

k=240\,\mathrm{N/m}

moves from a stretch of

+0.12\,\mathrm{m}

to its relaxed position. Find the work done by the spring.

Define the motion
The system is the spring and attached object, in a frame fixed to the support. Right is positive. The initial displacement is $x_i=+0.12\,\mathrm{m}$ and the final displacement is $x_f=0\,\mathrm{m}$ . The object moves left, in the same direction as the spring force.
Use the spring-work relationship
Work done by the spring equals the decrease in elastic potential energy. Substitute both endpoint displacements.
$W_s=\frac{1}{2}(240\,\mathrm{N/m})\left[(0.12\,\mathrm{m})^2-(0\,\mathrm{m})^2\right]$
Calculate and check
The work is positive because the spring’s force and the motion are both leftward. The units are $(\mathrm{N/m})(\mathrm{m^2})=\mathrm{N\,m}=\mathrm{J}$ . The spring loses stored energy as it returns to equilibrium.
$W_s=1.728\,\mathrm{J}\approx1.7\,\mathrm{J}$

Answer: The spring does

1.7\,\mathrm{J}

of positive work, to two significant figures.

Check: A positive result is reasonable because the spring releases stored energy while returning to its relaxed position.

Worked example

Energy stored by compressing a spring

A spring with

k=320\,\mathrm{N/m}

is compressed

0.050\,\mathrm{m}

from its relaxed position. Find the elastic potential energy stored. Assume the compression is performed slowly.

Set the system and known values
The system is the spring. Use a frame fixed to its support and choose right as positive. The compression is to the left, so $x_f=-0.050\,\mathrm{m}$ ; initially $x_i=0\,\mathrm{m}$ . The unknown is the increase in elastic potential energy.
Relate external work to stored energy
Because the spring is compressed slowly, the external agent’s work is stored as elastic potential energy. The square makes the stored energy positive for compression.
$\Delta E_{\mathrm{el}}=\frac{1}{2}(320\,\mathrm{N/m})\left[(-0.050\,\mathrm{m})^2-(0\,\mathrm{m})^2\right]$
Calculate and check
The units reduce to joules. The negative displacement does not make the energy negative because it is squared. The final energy is small, as expected for a small compression.
$\Delta E_{\mathrm{el}}=0.400\,\mathrm{J}$

Answer: The spring stores

0.400\,\mathrm{J}

of elastic potential energy.

Check: The energy is positive and has SI units of joules. A compression stores energy just as a stretch does.

Common mistakes and how to avoid them

Writing the spring force in the same direction as displacement.

Correction: Use the negative sign in Hooke’s law. The spring force points toward the relaxed position.

Using the final spring force times the full displacement to calculate work.

Correction: The force changes during the motion. Use the spring-work relationship based on the initial and final squared displacements.

Saying elastic potential energy is negative when the spring is compressed.

Correction: With zero energy at the relaxed position, elastic potential energy is proportional to

x^2

and is non-negative.

Confusing work by the spring with work by an external agent.

Correction: For a slow deformation, external work increases stored energy; work by the spring has the opposite sign.

Lesson summary

Hooke’s law says the spring force is proportional to displacement and opposite in direction.
A spring’s force changes as it stretches or compresses, so use the spring-work relationship rather than treating its force as constant.
Elastic potential energy is $\frac{1}{2}kx^2$ , measured in joules, and is non-negative for either stretching or compression.
Work done by the spring is the negative change in elastic potential energy. For a slow deformation, external work equals that change.

Check your understanding

Question 1

A spring is compressed to

x=-0.040\,\mathrm{m}

. What is true about its spring force and elastic potential energy?

The spring force points left, and the energy is negative.
The spring force points right, and the energy is positive.
The spring force points right, and the energy is negative.
The spring force is zero, and the energy is positive.

Show answer and explanation

The spring force points right, and the energy is positive.

With right positive, a negative displacement means compression to the left. The spring force points opposite the displacement, to the right. Elastic potential energy depends on

x^2

, so it is positive for a nonzero compression.

Question 2

A spring’s elastic potential energy increases by

0.60\,\mathrm{J}

during a slow stretch. What is the work done by the spring?

$+0.60\,\mathrm{J}$
$-0.60\,\mathrm{J}$
$0\,\mathrm{J}$
$+1.20\,\mathrm{J}$

Show answer and explanation

-0.60\,\mathrm{J}

Work done by the spring is the negative change in elastic potential energy. Therefore, when the stored energy increases by

0.60\,\mathrm{J}

, the spring does

-0.60\,\mathrm{J}

of work.

Question 3

A spring has

k=100\,\mathrm{N/m}

and is stretched from equilibrium to

0.10\,\mathrm{m}

. How much elastic potential energy is stored?

$0.10\,\mathrm{J}$
$0.50\,\mathrm{J}$
$1.0\,\mathrm{J}$
$10\,\mathrm{J}$

Show answer and explanation

0.50\,\mathrm{J}

Using

E_{\mathrm{el}}=\frac{1}{2}kx^2

, the energy is

\frac{1}{2}(100\,\mathrm{N/m})(0.10\,\mathrm{m})^2=0.50\,\mathrm{J}

. The units reduce to joules.

Key terms

Relaxed position: The spring’s position when it is neither stretched nor compressed.
Displacement: A change in position that includes direction; for a spring, it is measured from the relaxed position.
Spring constant: A measure of a spring’s stiffness, represented by $k$ and measured in newtons per metre.
Restoring force: A force that acts toward the position from which a system has been displaced.
Work: Energy transferred when a force acts through a displacement.
Elastic potential energy: Energy stored in a stretched or compressed spring.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C3.1. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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C3.1 · Relate Hooke’s law, work, and elastic potential energy

Ontario Grade 12 Physics

How a spring’s force and energy change as it stretches or compresses

What you will learn

1. Hooke’s law: force depends on displacement

2. Work and the spring’s changing force

3. Choosing the right work relationship

Finding the spring force

Work done by a spring while it returns

Energy stored by compressing a spring

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH4U

About this lesson and its review