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A.3.6 Elastic Potential Energy — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 37 of 63 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

A.3.6 Elastic Potential Energy — IB PHYSICS SL notes: Calculate the energy stored in a stretched or compressed spring using Hooke's law and force–extension graphs.Elastic potential energy = the energy stored in a material, such as a spring, when it is stretched or compressed.Elas…

  • Elastic deformation
  • Beyond the elastic limit
  • What the symbols mean
  • Doubling the extension
  • Compression
  • Unit check

Part 1 of 7Elastic deformation and Hooke's law

Elastic potential energy
Elastic potential energy the energy stored in a material, such as a spring, when it is stretched or compressed.
Elastic deformation
  • Elastic deformation the material returns to its shape when the force is removed, so the energy can be recovered.
Beyond the elastic limit
  • Beyond the elastic limit the deformation is permanent and part of the work is dissipated as thermal energy.
Hooke's law
  • F = kx, valid up to the limit of proportionality.
    What the symbols mean: F = force (N) · k = spring constant (N m−1) · x or e = extension (m)
  • F is the force in N, x the extension or compression from the natural length in m and k the spring constant in N m−1.
Graph
  • F against x is a straight line through the origin with gradient k, curving beyond the limit.

Part 2 of 7Energy stored

Work
  • Work the force rises from zero to kx, so the work equals the area under the force–extension graph, a triangle.
Equations
  • EH = ½kx2 = ½Fx.
  • EH is in J, k in N m−1 and x in m.
Doubling the extension
  • Doubling the extension it quadruples the stored energy.
Compression
  • Compression the same formula applies.
Unit check
  • N m−1 × m2 gives N m, a joule.

Part 3 of 7Graph interpretation

Area between extensions
  • Area between extensions the energy transferred in going from x1 to x2 is ½k(x22 − x12).
  • It is not ½k(x2 − x1)2.
Non-linear spring
  • Non-linear spring count squares and multiply by the energy of one square.
Rubber
  • Rubber loading and unloading curves differ, and the area between them is energy converted to thermal energy.
Offset in readings
  • Offset in readings measuring the total length shifts the intercept but not the gradient.

Part 4 of 7Energy conversions with springs

Energy conversions with springs
Launching a trolley
  • ½kx2 = ½mv2 if no energy is lost.
Wire breaking
  • Wire breaking the stored energy becomes kinetic energy, ½kx2 = ½mv2, so v ∝ x.
  • A greater extension at breaking gives a greater speed, which makes it dangerous.
Mass released from the unstretched position
  • Mass released from the unstretched position it falls a distance d before stopping, with mgd = ½kd2, so d = 2mg/k.
Equilibrium extension
  • Equilibrium extension it is only mg/k, which is why the mass oscillates.
Shock absorber
  • Shock absorber a spring absorbs energy when a car goes over a bump.

Part 5 of 7Springs in combination

Parallel
  • Parallel the load is shared and k = k1 + k2.
Series
  • Series the same force acts on each and the extensions add, so 1/k = 1/k1 + 1/k2.
Stored energy
  • Stored energy it is the sum of the energy in each spring in either arrangement.

Part 6 of 7Solving problems

1
Draw a sketch showing the natural length, the stretched or compressed length and the extension x.
2
Check that k is in N m−1 and x in m.
3
Apply EH = ½kx2.
4
For a spring supporting a mass, include the change in gravitational potential energy as well.
Tension and compression
  • Tension and compression the energy formula is identical for both.
Finding k
  • Finding k it is the gradient of an F–x graph.

Part 7 of 7More from the specification

More from the specification
Mechanical energy storage in an elastic system
  • Mechanical energy storage in an elastic system winding a tension spring provides an electromechanical means to run an auxiliary drive motor.
Elastic potential energy
  • Elastic potential energy energy stored in a stretched spring, E = one half k x squared, equal to the area under the force-extension graph.
Hooke's law
  • Hooke's law force is proportional to extension within the limit of proportionality.
Conservation of energy
  • Conservation of energy in an arrow fired from a bow, stored elastic energy becomes kinetic energy and then gravitational energy at height.
Efficiency
  • Efficiency useful output divided by input; wasted energy goes to thermal stores.
Power
  • Power rate of energy transfer, which links mechanical work to time in horizontal or vertical motion.
Energy conservation
  • Mgh = ½Mv2 + ½Iω2.
Resultant force
  • Another name for net force.
Final speed
  • Speed at the end, v
Efficiency
  • Useful output divided by total input.
Friction
  • Force parallel to the surface opposing relative motion or its tendency.
Density
  • Mass per unit volume.
Component
  • Part of a vector along one axis.
Wasted energy
  • Energy transferred to stores that are not useful, usually thermal in the surroundings.
Mechanical energy
  • Sum of kinetic and potential energy.
Energy density
  • The energy available per unit mass or area of a resource.
Sankey diagram
  • Diagram where arrow width shows energy amount.
W
Fs = 25.6 × 20 = 512 J
What the symbols mean: W = work done = energy transferred (J) · F = force (N) · d, s or x = distance moved in the direction of the force (m)

Key terms

TermMeaning
Elastic potential energyEnergy stored in a stretched or compressed elastic material, ½kx2
Spring constant (k)Force per unit extension, in N m−1
ExtensionIncrease in length from the natural length
Limit of proportionalityPoint beyond which force is not proportional to extension
Elastic limitPoint beyond which deformation is permanent
HysteresisDifference between loading and unloading curves of a material

Avoid these mistakes

1
Using the total length instead of the extension — x is measured from the natural length.
2
Dropping the ½ in ½kx2 — the force rises from zero.
3
Squaring only part of the expression — the extension is squared, not the constant.
4
Finding the energy between two extensions as ½k(Δx)2 — use the difference of squares.
5
Applying the formula beyond the limit of proportionality — the graph is no longer a straight line.

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