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Physics · Forces

Forces and elasticity

Describe elastic and inelastic deformation, use \(F = ke\) and \(E_e = \tfrac{1}{2}ke^2\), and investigate the relationship between force and extension (Required Practical 6).

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    What is a spring?

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    A coiled wire that can be stretched or compressed

  2. 2

    What does proportional mean?

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    Doubling one doubles the other

  3. 3

    Convert 6 cm to m.

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    0.06 m

  4. 4

    What is elastic potential energy?

    Show answerHide answer

    Energy stored in a stretched spring

  5. 5

    What is the unit of force?

    Show answerHide answer

    Newton

Learning Objectives

  1. 1Explain why more than one force is needed to stretch, bend or compress an object.
  2. 2Describe elastic and inelastic deformation.
  3. 3Recall and apply \(F = ke\) and \(E_e = \tfrac{1}{2}ke^2\).
  4. 4Interpret force-extension graphs and describe Required Practical 6.

HOOKE'S LAW

force \(=\) spring constant \(\times\) extension \(F = ke\)

The extension of an elastic object is directly proportional to the force applied, provided the limit of proportionality is not exceeded. The same applies to compression.

Required Practical 6: Method

Investigate force and extension.

  1. 1 Measure the original length

    With no masses on the spring.
  2. 2 Add a mass

    Record the force (weight = mg) and the new length.
  3. 3 Calculate the extension

    New length minus original length.
  4. 4 Repeat

    Add masses one at a time, keeping below the limit of proportionality at first.
  5. 5 Plot

    Force (y-axis) against extension (x-axis).
  6. 6 Gradient

    Gradient of the straight part = spring constant k.

Spring Constant

A force of 3.0 N stretches a spring by 6.0 cm (within the limit of proportionality). Calculate the spring constant.

Show the solutionHide the solution
  1. 1 Convert \(6.0\) cm \(= 0.060\) m
  2. 2 Rearrange \(k = F \div e\)
  3. 3 Substitute \(k = 3.0 \div 0.060\)
  4. 4 Answer \(k = 50\) N/m

Answer50 N/m

Elastic Potential Energy

Calculate the energy stored in the spring when the extension is 0.040 m and k = 50 N/m.

Show the solutionHide the solution
  1. 1 Write the equation \(E_e = \tfrac{1}{2}ke^2\)
  2. 2 Substitute \(E_e = 0.5 \times 50 \times 0.040^2\)
  3. 3 Answer \(E_e = 0.040\) J

Answer0.040 J

Elastic or Inelastic?

Elastic deformation

  • The object returns to its original shape when the force is removed.
  • Happens within the limit of proportionality.

Inelastic deformation

  • The object does not return to its original shape.
  • Happens beyond the limit of proportionality.

Key Ideas

Learn these.

  • Two forces

    To stretch, bend or compress a stationary object, more than one force is needed (for example pull at both ends).

  • Work done

    The work done on an elastic object equals the elastic potential energy stored, provided it is not inelastically deformed.

  • Linear

    A straight line through the origin: proportional.

  • Non-linear

    A curve: not proportional.

Spring Test

A spring stretches 4.0 cm when a 2.0 N force is applied. (a) Calculate k in N/m. (b) What force gives an extension of 10 cm if the limit is not exceeded? (c) What energy is stored at 10 cm?

1. Convert cm to m.

2. Use F = ke and Ee = ½ke².

A good answer shows: (a) 50 N/m. (b) F = 50 × 0.10 = 5.0 N. (c) E = 0.5 × 50 × 0.10² = 0.25 J.

Can I...?

  1. 1State F = ke.
  2. 2Convert cm to m.
  3. 3Calculate a spring constant.
  4. 4Calculate stored energy.
  5. 5Describe elastic and inelastic deformation.
  6. 6Read a force-extension graph.
  7. 7Describe Required Practical 6.
  8. 8Find the gradient.

Summary & Exam Focus

  • \(F = ke\).
  • \(E_e = \tfrac{1}{2}ke^2\).
  • Beyond the limit of proportionality: non-linear, may not return.
  • Spring constant = gradient of the linear part.

Exam focus

A spring has a spring constant of 50 N/m. Calculate the force needed to stretch it by 0.10 m. (2 marks) (2 marks)

Use F = ke.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Extension
The increase in length of a spring.
Spring constant
The force needed per metre of extension (N/m).
Limit of proportionality
The point beyond which force and extension are no longer proportional.
Elastic deformation
A change of shape that is reversed when the force is removed.
Inelastic deformation
A permanent change of shape.
Linear
Following a straight line.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Calculate 2 marks

    A spring has a spring constant of 50 N/m. Calculate the force needed to stretch the spring by 0.10 m. Use the equation: force = spring constant × extension

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    Model answer

    \(50 \times 0.10 = 5.0\) N

    Mark scheme

    • Correct substitution — 1 mark
    • 5.0 N — 1 mark
  2. Question 2 Use the graph 5 marks

    A student investigates a spring. The graph shows how the force changes with extension. (a) State the extension at the limit of proportionality. (b) Calculate the spring constant of the spring. (c) Calculate the elastic potential energy stored when the extension is 0.040 m.

    A force-extension graph that is straight to 6 centimetres then curves.
    Show answerHide answer

    Model answer

    (a) About 6 cm. (b) \(k = 3.0 \div 0.060 = 50\) N/m. (c) \(E_e = 0.5 \times 50 \times 0.040^2 = 0.040\) J

    Mark scheme

    • 6 cm (accept 5.5 to 6.5) — 1 mark
    • Uses 3.0 N at 0.060 m — 1 mark
    • 50 N/m — 1 mark
    • Correct substitution — 1 mark
    • 0.040 J — 1 mark
  3. Question 3 Describe 2 marks

    Describe the difference between elastic deformation and inelastic deformation of a spring.

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    Model answer

    In elastic deformation the spring returns to its original length when the force is removed; in inelastic deformation it does not and is permanently changed.

    Mark scheme

    • Elastic: returns to original shape — 1 mark
    • Inelastic: does not return — 1 mark
  4. Question 4 Describe 6 marks

    Describe how to investigate the relationship between the force applied to a spring and the extension of the spring. Your answer should include the apparatus, the measurements, and how you would use them.

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    Model answer

    See levels-of-response scheme.

    Mark scheme

    • Level 3 (5 to 6 marks): a complete method with stand, ruler and masses, original length measured, several forces (weights) applied with the new length each time, extension calculated, graph plotted and spring constant from the gradient, with a precaution (eye level, avoid overstretching) — 5 to 6 marks
    • Level 2 (3 to 4 marks): a method covering force and extension with some detail missing — 3 to 4 marks
    • Level 1 (1 to 2 marks): simple statements about adding masses and measuring length — 1 to 2 marks
  5. Question 5 Explain 2 marks

    Explain why two forces are needed to stretch a spring that is fixed to a wall.

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    Model answer

    The wall applies a force in the opposite direction to the pulling force, so the spring is stretched; with only one force the object would accelerate rather than change shape.

    Mark scheme

    • Forces act at both ends — 1 mark
    • Equal and opposite forces — 1 mark

Quick check

  1. Extension is...

    1. Aoriginal length
    2. Bnew length minus original length
    3. Cthe force
    4. Dthe mass
    Show answerHide answer

    B: new length minus original length

    It is the increase in length.

  2. A 4 N force gives a 0.08 m extension. k =

    1. A0.32 N/m
    2. B5 N/m
    3. C50 N/m
    4. D320 N/m
    Show answerHide answer

    C: 50 N/m

    4 ÷ 0.08 = 50.

  3. A straight line force-extension graph through the origin shows...

    1. Ainverse proportion
    2. Bno relationship
    3. Cinelastic behaviour
    4. Ddirect proportion
    Show answerHide answer

    D: direct proportion

    F ∝ e.

  4. Beyond the limit of proportionality the spring...

    1. Amay not return to its original length
    2. Balways returns
    3. Chas zero extension
    4. Dstops being a spring
    Show answerHide answer

    A: may not return to its original length

    It is inelastically deformed.

  5. The gradient of the linear section is the...

    1. Aextension
    2. Bspring constant
    3. Cmass
    4. Dwork done
    Show answerHide answer

    B: spring constant

    F ÷ e.

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