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Hooke's law and springs Double the force on a spring: is it twice as long?

Foundation and Higher · AQA · 45 minutes

Students predict how long a spring gets when the force on it doubles, then load a live spring themselves and watch its force–extension graph grow, straight and then curving. The required practical, the energy stored and rearranging follow. Slides to present, a vote that shows who's stuck, practice and homework.

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A spring is 10 cm long with nothing on it. Across: the force pulling it, in newtons; up: its length, in cm; a: the centimetres each newton adds. With a = 2, 2 N makes it 14 cm long. Does twice the force make it twice as long? Move a and compare 2 N with 4 N.

246810

y = 10 + 2x

The idea in one minute

The force stretching a spring is proportional to its extension, the extra length, not to its length: F = k e, with the force in newtons, the extension in metres and the spring constant k in newtons per metre (N/m). A stiffer spring has a bigger k. Hung masses pull with their weight, W = m g, so grams become kilograms and then newtons first (g = 9.8 N/kg). The force–extension graph is a straight line through the origin, its gradient k, until the limit of proportionality; past it the line curves. A spring that goes back to its length when unloaded was deformed elastically; one that stays stretched, inelastically. Stretching a spring stores energy: Ee = ½ k e².

The lesson starts with a prediction before anyone explains: a spring 14 cm long with 2 N, how long with 4 N? Students then load a live spring (a simulation, not a substitute for the practical), work through a student's results from the required practical, and turn every length into an extension in metres. It is for Foundation and Higher; the energy stored leans towards Higher, and rearranging it is the stretch.

By the end, students can

  • use F = k e with the extension in metres, and rearrange it for k or e
  • turn hung masses into a force with W = m g
  • describe the force–extension graph, its gradient and the limit of proportionality
  • tell elastic from inelastic deformation, and describe the required practical
  • work out the energy stored in a stretched spring with Ee = ½ k e²

Key words

  • extension
  • spring constant (N/m)
  • Hooke's law
  • limit of proportionality
  • elastic deformation
  • inelastic deformation
  • elastic limit
  • force–extension graph
  • elastic potential energy

Before this lesson

Students should know that weight = mass × g (W = m g, g = 9.8 N/kg), turn centimetres into metres and rearrange a formula such as F = k e.

Taught in Newton's laws of motion.

The lesson, step by step

45 minutes in total. The stretch is optional.

  1. 1 · Think5 min

    How long with twice the force?

    A spring: 10 cm long with nothing on it

    2 N hung on it: now it is 14 cm long

    4 N instead: how long is it now?

    A: 28 cm

    B: 18 cm

    C: 16 cm

    Vote first. Then say why.

    Write here

    5 min 1 · Hook · students think · the slides say “Think”

    How long with twice the force?

    A spring 10 cm long is 14 cm long with 2 N: students vote on its length with 4 N before anyone explains.

    Say

    “This spring is 10 cm long with nothing on it. Hang 2 N on it and it is 14 cm long. Hang 4 N on it instead: how long is it? 28 cm, 18 cm or 16 cm?”

    Ask
    • What does the 2 N change: the whole length, or the extra length?
    • What would 0 N give? Does your answer fit that?
    Watch for

    Doubling the whole length (28 cm): keep that answer on the board, the next step needs it.

    For the teacher

    Give a minute of silent thinking, then vote. Many double the length: 28 cm. Some add 2 cm for the extra 2 N: 16 cm.

    Don't explain yet. Only the stretch doubles: 4 cm becomes 8 cm, so the spring is 18 cm long. That is the next step: extension, not length.

    Take a vote with Ask the class: 28 cm, 18 cm or 16 cm. Keep the votes on the board for the explain step.

    Shorter: Vote once, no discussion (−2 min).

  2. 2 · Watch12 min

    Hooke's law: F = k e

    Force (N) = spring constant (N/m) × extension (m)

    Extension: stretched length − original length

    The hook: 2 N gave e = 14 − 10 = 4 cm

    4 N: twice the extension, 8 cm: 10 + 8 = 18 cm long

    k: the newtons for each metre of stretch

    A stiffer spring has a bigger k

    Extension, never the length

    2 · Watch

    F = k e on a graph

    x: extension e (cm)

    y: force F (N)

    A straight line through (0, 0)

    Gradient = F ÷ e = k

    e in metres: 4 cm = 0.04 m

    2 N ÷ 0.04 m = 50 N/m

    Move k: which line is stiffer?

    2 · Watch

    Masses hung on a spring

    Their weight pulls it: W = m g, g = 9.8 N/kg

    Grams to kilograms: ÷ 1000. 100 g = 0.1 kg

    100 g: W = 0.1 × 9.8 = 0.98 N

    Centimetres to metres: ÷ 100

    500 g on k = 98 N/m: W = 0.5 × 9.8 = 4.9 N

    e = 4.9 ÷ 98 = 0.05 m = 5 cm

    Newtons and metres before F = k e

    2 · Watch

    The limit of proportionality

    x: extension (cm), y: force (N)

    Straight: F ∝ e, Hooke's law

    The dot: limit of proportionality

    Past it, the line curves:

    each newton stretches it more

    Past the elastic limit: stays longer

    F = k e on the straight part only

    2 · Watch

    Elastic or inelastic?

    Stretch, bend or squash: it takes two forces

    One force alone would just move the object

    Elastic: back to its shape when the forces go

    A spring below its limit, a rubber band, a diving board

    Inelastic: it stays out of shape

    Modelling clay, a bent paper clip, a ruined spring

    Does it go back? Then it's elastic.

    2 · Watch

    Try it: double the load

    300 g on a 25 N/m spring: 11.8 cm

    Predict 600 g, then set it

    Now k = 50 N/m: predict, then check

    k = 20 N/m: add masses past 800 g

    Predict first: twice F, twice e?

    SpringsWeight 2.94 N · extension 11.8 cm (F = k e) · energy stored 0.173 J

    2 · Watch

    Energy stored in a spring

    Stretching it does work on it,

    stored as elastic potential energy

    Ee = ½ k e², e in metres

    k = 50 N/m, e = 8 cm = 0.08 m:

    ½ × 50 × 0.08² = 0.16 J

    The shaded area under the line

    Metres first, then square e

    12 min 2 · Explain · you explain · the slides say “Watch”

    Hooke's law, its graph and its limit

    F = k e with the extension, newtons and metres first; the force–extension graph and its limit; elastic and inelastic; a live spring; then the energy stored.

    Say

    “The force is proportional to the extension, the extra length, not to the length. Newtons and metres before anything else.”

    Ask
    • The spring was 10 cm long and is now 14 cm long. What is its extension?
    • Which line is the stiffer spring?
    • When does F = k e stop working?
    Watch for

    Putting the stretched length into F = k e, or the extension in centimetres: ask how much longer, and in metres, every time.

    The hook, worked

    2 N: e = 14 − 10 = 4 cm

    4 N: twice the force, twice the extension: 8 cm

    Length = 10 + 8 = 18 cm; k = 2 ÷ 0.04 = 50 N/m

    Energy stored

    k = 50 N/m, stretched 8 cm = 0.08 m

    Ee = ½ × 50 × 0.08² = 0.16 J

    For the teacher

    Hooke's law: the force on a spring is proportional to its extension, F = k e, F in newtons, e in metres, k the spring constant in N/m: the force for each metre of stretch. A stiffer spring has a bigger k. Extension = stretched length − original length. The hook: 2 N gave 4 cm, so 4 N gives 8 cm, and the spring is 18 cm long. Its k is 2 ÷ 0.04 = 50 N/m.

    Hung masses: the force is their weight, W = m g, with m in kilograms and g = 9.8 N/kg. 100 g gives 0.98 N. Centimetres to metres: divide by 100.

    The force–extension graph (force up, extension along) is a straight line through the origin while the law holds; its gradient is k. Past the limit of proportionality it curves: each extra newton stretches it more. The elastic limit is close by: past it, the spring stays longer when the load comes off.

    Changing a shape (stretching, bending, squashing) needs more than one force: one alone would just move the object. Elastic deformation: it goes back to its shape when the forces are removed. Inelastic: it doesn't. F = k e works for squashing a spring too, e being how much shorter it is.

    The live spring: 300 g on 25 N/m stretches it 11.8 cm. Ask for a prediction for 600 g (23.5 cm), then for k = 50 N/m (11.8 cm again: twice as stiff), then set k = 20 N/m and add masses past 800 g: the graph curves and the readout says it won't go back. The simulation puts both limits at the same extension, 40 cm; real springs have them close, not always the same.

    Energy stored: stretching a spring does work on it, stored as elastic potential energy, Ee = ½ k e², e in metres, as long as it isn't deformed inelastically. It is the area under the straight line. Twice the extension stores four times the energy. The equation is given in the exam; rearranging it is the harder part.

    On the Try it slide, ask for a prediction before each move: 600 g, then k = 50 N/m, then k = 20 N/m with masses past 800 g.

    Shorter: Skip the elastic-or-inelastic slide and the energy slide's area; say each in a line (−4 min).

  3. 3 · Your turn12 min

    The required practical

    Clamp a spring above a vertical ruler; fix a pointer

    Read its length with nothing on it: the original length

    Add 100 g masses one at a time; read the pointer

    Read at eye level, once the masses are still

    Extension = length now − original length

    Plot force (up) against extension (along)

    Safety: clamp it down, feet clear of the masses

    3 · Your turn

    A student's results

    Unloaded: 6.0 cm long

    100 g: 10.9 cm; 200 g: 15.8 cm

    300 g: 20.7 cm; 400 g: 25.6 cm

    x: extension (cm), y: force (N)

    Straight, through (0, 0)?

    Its gradient is k: in N/m?

    Open the Practical quiz.

    3 · Your turn

    Worked, then yours

    I do: k = 60 N/m, e = 5 cm: F = 60 × 0.05 = 3 N

    You do: k = 45 N/m, e = 8 cm: the force?

    I do: 200 g, k = 49 N/m: e = 1.96 ÷ 49 = 0.04 m

    You do: 450 g, k = 90 N/m: the extension?

    I do: k = 120 N/m, e = 0.05 m: Ee = 0.15 J

    You do: k = 300 N/m, e = 6 cm: the energy stored?

    Metres and newtons first.

    Write here

    Its answers' slide comes with your copy.

    12 min 3 · Your turn · students write

    The practical, then yours

    The required practical step by step, a student's results to turn into extensions and a spring constant, then twins of worked examples.

    Say

    “This is the required practical. Read the length, take away the original length, every time.”

    Ask
    • Why read the pointer with your eye level with it?
    • If Hooke's law holds, what will the graph look like?
    Watch for

    Writing the ruler's reading as the extension.

    F = k e: a force

    k = 60 N/m, stretched 5 cm

    e = 5 ÷ 100 = 0.05 m

    F = 60 × 0.05 = 3 N

    Then theirs

    k = 45 N/m, stretched 8 cm. The force?

    Mass first, then e = F ÷ k

    200 g hangs on a spring of k = 49 N/m

    W = 0.2 × 9.8 = 1.96 N

    e = 1.96 ÷ 49 = 0.04 m = 4 cm

    Then theirs

    450 g hangs on a spring of k = 90 N/m. Its extension?

    Energy stored

    k = 120 N/m, stretched 0.05 m

    Ee = ½ × 120 × 0.05² = 0.15 J

    Then theirs

    k = 300 N/m, stretched 6 cm. The energy stored?

    A student hung a spring from a clamp beside a vertical ruler, with a pointer at its bottom, and added 100 g masses. With nothing on it the spring was 6.0 cm long; with 100 g, 200 g, 300 g and 400 g it was 10.9, 15.8, 20.7 and 25.6 cm long. Use g = 9.8 N/kg.

    1. With 300 g on it, what was the spring's extension, in cm?
    2. What force does 300 g pull the spring with, in newtons?
    3. Why does the student read the pointer with their eye level with it?
      1. So the reading isn't shifted by looking from above or below
      2. So the spring stretches further
      3. So the masses hang still
    4. With 400 g the force is 3.92 N. Use that reading to find the spring constant, in N/m.
    5. Plotted with force up and extension along, the readings make a straight line through the ___. Its gradient is the ___.
    6. Predict the spring's extension with 250 g on it, in cm.
    7. With 600 g the spring was 40.0 cm long, and with all the masses off it stayed 7.5 cm long. What happened?
      1. It went past its limit: it stretched more than F = k e says and was deformed inelastically
      2. It obeyed Hooke's law all the way and was deformed elastically
      3. Nothing: a spring always comes back a little longer
    For the teacher

    The practical: clamp a spring above a vertical ruler, with a pointer fixed to its bottom. Read its length with nothing on it. Add 100 g masses one at a time, reading the pointer at eye level once the masses are still. Extension = length − original length. Plot force (the weight, W = m g) up against extension along: a straight line through the origin, its gradient k with the extension in metres. Safety: clamp the stand to the bench, goggles on, feet clear of the masses.

    The quiz gives a student's results: 6.0 cm unloaded, then 10.9, 15.8, 20.7 and 25.6 cm with 100 to 400 g. Students turn lengths into extensions, masses into newtons, find k and predict a reading; the last question is a reading past the limit.

    Work the examples aloud while the slower ones finish the quiz; the twins check they turn centimetres into metres, grams into newtons, and square the extension.

    Students open the Practical quiz on their own devices. With springs and masses in the room, they take their own readings first. Open the answers slide once most have done the twins.

    Shorter: The practical quiz's first four questions and the first pair only (−4 min).

    Longer: Each student writes a spring question for a partner, with a mass, k and the extension in cm, and works it out first (+3 min).

  4. 4 · Vote and practise12 min

    Vote first

    A spring is 12 cm long. A weight stretches it

    to 18 cm long. k = 50 N/m. What is the weight?

    A: 9 N

    B: 3 N

    C: 300 N

    D: 6 N

    Choose on your own, then say why.

    4 · Vote and practise

    Practise: show your working

    1. k = 40 N/m, e = 0.05 m: the force?

    2. 6 N on a spring of k = 120 N/m: the extension?

    3. 4.5 N stretches a spring 9 cm: k in N/m?

    4. 250 g on k = 35 N/m: the extension in cm?

    5. Which is elastic deformation?

    Stuck on 4? Weight first: W = m g

    Write here

    4 · Vote and practise

    Reason it out

    6. The graph curves: what does that show?

    7. Why does stretching need two forces?

    8. k = 80 N/m, e = 0.15 m: the energy stored?

    9. k = 200 N/m, e = 4 cm: the energy stored?

    10. 0.5 J stored at 10 cm: k?

    Stretch: 25 N/m, limit at 40 cm: the largest mass?

    Square e, in metres, then halve.

    12 min 4 · Who's stuck? · students vote · the slides say “Vote and practise”

    Vote, then practise

    One question shows who puts a length or centimetres into F = k e; then practice in levels, and the largest load before the limit as the stretch.

    Say

    “Choose an answer on your own. Nobody talks until everyone has chosen.”

    Ask
    • How much longer did it get? And in metres?
    Watch for

    Using 18 cm or 12 cm, or 6 without turning it into metres.

    Show your working on paper. Use g = 9.8 N/kg. Turn centimetres into metres and grams into newtons first. The stretch is optional.

    The question that shows who's stuck

    1. A spring is 12 cm long. A weight hung on it stretches it to 18 cm long. Its spring constant is 50 N/m. What is the weight?
      1. 9 N
      2. 3 N
      3. 300 N
      4. 6 N
    In your Classroom, as they answer (an example)
    • A9
    • B10
    • C5
    • D4

    Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.

    1. A spring has a spring constant of 40 N/m. What force stretches it by 0.05 m, in newtons?
    2. A force of 6 N stretches a spring of spring constant 120 N/m. What is its extension, in metres?
    3. A force of 4.5 N stretches a spring by 9 cm. What is its spring constant, in N/m?
    4. A 250 g mass hangs on a spring of spring constant 35 N/m. What is the spring's extension, in cm?
    5. Which of these is elastic deformation?
      1. A ruler bent a little that springs back straight
      2. A paper clip bent into a new shape
      3. A lump of modelling clay squashed flat
      4. A spring stretched so far it stays longer
    6. A spring's force–extension graph is a straight line through the origin, then it curves. What does the curve show?
      1. The spring has gone past its limit of proportionality: F = k e no longer works
      2. The spring has become stiffer
      3. The spring constant has doubled
    7. A spring hangs from a clamp with a mass on it. Why does stretching it need two forces?
      1. One force alone would just move the spring: the clamp pulls up on its top while the weight pulls down on its bottom
      2. One force stretches it and the other makes it go back afterwards
      3. It doesn't: the weight stretches it and the clamp does nothing
    8. A spring of spring constant 80 N/m is stretched by 0.15 m. How much energy does it store, in joules?
    9. A spring of spring constant 200 N/m is stretched by 4 cm. How much energy does it store, in joules?
    10. A spring stretched by 10 cm stores 0.5 J. What is its spring constant, in N/m?
    11. Stretch: a spring of spring constant 25 N/m reaches its limit of proportionality at an extension of 40 cm. What is the largest mass, in grams, it can hold and still obey Hooke's law? Then try it on the live spring.

    Support · Write three lines before any F = k e: the extension (stretched length − original length), the extension in metres, and the force in newtons (W = m g for a mass).

    For the teacher

    Everyone chooses before anyone explains. Then ask in turn: how long was it? How much longer is it now? In metres? Times k?

    Questions 1–3 build fluency; 4–5 change the look (grams, elastic or not); 6–7 ask for reasons; 8–10 are the energy stored, 10 rearranged. The stretch finds the largest load before the limit; check it on the live spring afterwards.

    Students show their working on paper or the board; the quiz keeps their final answers.

    If many chose 9 N or 6 N, draw the spring beside a ruler and mark the original length before they practise. If many chose 300 N, ask what unit k is in.

    Shorter: After the vote, questions 1–5 only (−5 min).

    Longer: The stretch: the largest load before the limit, then check it on the live spring (+8 min).

  5. 5 · Show what you know4 min

    How long now?

    A spring is 8.0 cm long

    With a 2.0 N weight it is 13.0 cm long

    1. What is its spring constant?

    2. How long is it with 3.0 N?

    Extension first, in metres.

    Write here

    4 min 5 · Exit · students show · the slides say “Show what you know”

    How long now?

    Lengths, not extensions: shows who takes the extension and turns it into metres.

    Say

    “Extension first, then metres.”

    Watch for

    Giving 7.5 cm as the length, or 0.4 N/m for k.

    Write the extension in metres before using F = k e.

    1. A spring is 8.0 cm long. With a 2.0 N weight on it, it is 13.0 cm long. What is its spring constant, in N/m?
    2. How long is the spring with a 3.0 N weight on it, in cm?
    For the teacher

    A spring is 8.0 cm long. With a 2.0 N weight on it, it is 13.0 cm long.

    What is its spring constant? How long is it with 3.0 N?

    Those who used a length or centimetres need it again next lesson; set the homework.

    Longer: How much energy does it store with 3.0 N on it? (+4 min).

One to one: before any formula, ask the student how much longer the spring got, and then in metres; let them drive the live spring while you only ask what they predict before each move.

Homework: Hooke's law and springs

Use g = 9.8 N/kg. Write the extension in metres and the force in newtons before using a formula. Questions 4 and 5 are about the energy stored.

  1. A spring has a spring constant of 30 N/m. What force stretches it by 0.12 m?
  2. A 500 g mass hangs on a spring of spring constant 70 N/m. Find the mass's weight, then the spring's extension in cm.
  3. A spring 15 cm long is 21 cm long when 1.8 N hangs on it. Find its spring constant.
  4. A spring of spring constant 250 N/m is stretched by 8.0 cm. How much energy does it store?
  5. A spring of spring constant 400 N/m stores 2.0 J. How far is it stretched, in cm?
  6. Describe how you would find a spring's spring constant in the lab: what you measure, how, and what you plot.
  7. A student stretches a spring a little, and a piece of modelling clay, then lets go of each. Describe what happens to each, using the words elastic and inelastic.

Print the worksheet with working space and homework. Teacher answers come with your Library copy.

Teaching hooke's law and springs: common questions

Is Hooke's law on Foundation and Higher, and in combined science?

In AQA GCSE Physics (8463) forces and elasticity is for both tiers, and AQA's combined science (8464) has it too, with the same required practical on force and extension. The whole lesson suits both tiers; rearranging Ee = ½ k e² leans towards Higher.

What is the difference between the limit of proportionality and the elastic limit?

Up to the limit of proportionality, the extension is proportional to the force and the graph is a straight line. Up to the elastic limit, the spring goes back to its own length when the load comes off. For a spring they are close together; the live spring puts both at an extension of 40 cm, so it shows one limit.

How do students find the spring constant from the required practical?

Plot the force (the weight of the masses, W = m g) up and the extension along, draw a line of best fit through the straight part, and find its gradient with the extension in metres: that is k in N/m. A gradient worked out in centimetres comes out a hundred times too small. If the extension is plotted up instead, k is one over the gradient.

Should students use g = 9.8 N/kg or 10 N/kg?

Use what the question gives. AQA gives the value of g in any calculation, usually 9.8 N/kg, and the lesson uses 9.8 throughout, as the live spring does: 100 g pulls with 0.98 N.

What does the live spring show, and does it replace the practical?

It hangs 0 to 1000 g on a spring of 10 to 100 N/m beside a ruler, plots a reading for each 100 g on a force–extension graph and says the weight, the extension and the energy stored. At 20 N/m or less the heaviest loads take it past its limit, and the graph curves. It is a model that shows what to expect; it does not replace taking real readings.

Can I use the lesson without creating an account?

Yes. Read the lesson, print the worksheet and open its slides, with the live spring, without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.

Curriculum reference: AQA GCSE Physics 8463: 4.5.3 (forces and elasticity: elastic and inelastic deformation, F = k e, the limit of proportionality, force–extension graphs, the work done in stretching a spring, Ee = ½ k e²), with 4.5.1.3 (weight, W = m g) and required practical activity 6 (force and extension).

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