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A.2.8 Hooke’s Law — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 19 of 63 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

A.2.8 Hooke's Law — IB PHYSICS SL notes: State Hooke's law, use the spring constant, interpret force–extension graphs and calculate elastic energy.Hooke's law — the extension of a material is directly proportional to the applied force up to the limit of proportionality.It applies…

  • Hooke's law region
  • Beyond proportionality
  • Beyond the elastic limit
  • Rubber band
  • Swapped axes
  • Extension, not length

Part 1 of 7The law

Hooke's law
  • Hooke's law the extension of a material is directly proportional to the applied force up to the limit of proportionality.
  • It applies to any elastic material such as a spring, wire or bungee rope.
Equation
  • FH = −kx.
  • FH is the elastic restoring force in N.
  • k is the spring constant in N m−1.
  • x is the extension in m, and the minus sign shows the force opposes the displacement.
Spring constant
  • Spring constant it measures stiffness; a larger k means a stiffer spring.
Extension
  • Extension extension = stretched length − unstretched length, and compression is a decrease in length.

Part 2 of 7Force–extension graphs

Hooke's law region
  • Hooke's law region a straight line through the origin whose gradient is k.
Beyond proportionality
  • Beyond proportionality the line curves and the extension grows faster than the force.
Beyond the elastic limit
  • Beyond the elastic limit the material does not return to its original length and keeps a permanent extension.
Rubber band
  • Rubber band loading and unloading curves differ (hysteresis) and Hooke's law is not obeyed.
Swapped axes
  • Swapped axes if extension is on the y-axis and force on the x-axis, the gradient is 1/k.
Extension, not length
  • Extension, not length use the extension, not the total length, in the equation.

Part 3 of 7Elastic potential energy

Work done
  • Work done the area under the force–extension graph is the work done in stretching.
Stored energy
  • EH = ½kx2 = ½Fx = F2/(2k).
  • The factor ½ appears because the force rises from zero to F, so the average force is F/2.
Doubling the extension
  • Doubling the extension it quadruples the stored energy.
Release
  • Release for a spring that obeys Hooke's law the energy can become kinetic energy, as in a spring gun.

Part 4 of 7Springs in combination

Parallel
  • Parallel the load is shared, k = k1 + k2, so the combination is stiffer.
Series
  • Series each carries the full force and the extensions add, 1/k = 1/k1 + 1/k2, so it is softer.
Cut spring
  • Cut spring half the length gives twice the stiffness.

Part 5 of 7Testing Hooke's law

1
Hang masses from a spring and measure its length with a ruler at eye level to avoid parallax.
2
Calculate the extension as length minus natural length.
3
Allow the spring to settle before each reading.
4
Plot weight against extension and draw a best-fit line through the origin.
5
Take the gradient as the spring constant k.

Part 6 of 7Using the law

Hanging mass at rest
  • Hanging mass at rest mg = kx gives the equilibrium extension.
Energy method
  • Energy method a mass released from a compressed spring on a smooth surface has ½mv2 = ½kx2, so v = x√(k/m).
Vertical spring
  • Vertical spring include the change in gravitational potential energy as well as the elastic energy.
Measuring x
  • Measuring x measure it from the natural length, not from the equilibrium position of a hanging mass.
Applications
  • Applications spring balances, vehicle suspensions and pens.
Link
  • Link the restoring force F = −kx leads to simple harmonic motion with period 2π√(m/k).

Part 7 of 7More from the specification

1
Newton's First Law of Motion
  • Newton's first law states that an object continues in its state of rest or uniform velocity in a straight line unless acted upon by a non-zero resultant external force.
2
Newton's Second Law and Momentum
  • Newton's second law states that the resultant force acting on a body is directly proportional to the rate of change of linear momentum.
    • (F = dp/dt = ma for constant mass)
3
Frictional Opposing Forces
  • Friction is a contact force that always acts parallel to interface surfaces in a direction opposing relative motion, with maximum static friction given by Ff ≤ μ_s R.
4
Newton's Third Law Pairs
  • When body A exerts a force on body B, body B exerts an equal and opposite force on body A (FAB = -FBA), acting on different bodies with the same nature.
5
Angular speed
  • Magnitude of the angular velocity.
6
Linear speed
  • The distance travelled along the circle per unit time, v = ωr.
    What the symbols mean: v = speed (m s−1) · ω = angular speed (rad s−1) · r = radius (m) · T = period (s)
7
Normal reaction
  • The contact force perpendicular to a surface.
8
Reaction force
  • Force from a support pushing up on an object.
9
Contact force
  • Force requiring touching objects.
10
Fluid resistance
  • Drag force from a fluid opposing motion.
11
Collision
  • Short interaction with large internal forces.
12
Linear
  • A graph that is a straight line through the origin, so resistance is constant.
13
Impulse
  • J = FΔt = Δp.
    What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s−1); FΔt is the impulse (N s)
14
Radius
  • Distance from the centre to the circular path.
15
Normal
  • Line at 90° to a surface.
16
Buoyancy
  • Another name for upthrust.
17
Centripetal force
  • Net force toward the centre, mv2/r.
18
Static friction
  • Friction between surfaces at rest relative to each other; fs ≤ μsFN.
19
Momentum
  • P = mv, a vector.
20
Resultant
  • Vector sum of two or more vectors.
21
Velocity
  • Rate of change of displacement.
22
Acceleration
  • Rate of change of velocity.
23
Friction
  • Force parallel to the surface opposing relative motion or its tendency.
24
Newton
  • Unit of force, kg m s-2.

Key terms

TermMeaning
Hooke's lawExtension is proportional to the applied force up to the limit of proportionality
Spring constant (k)Force per unit extension, in N m−1; a measure of stiffness
ExtensionIncrease in length from the unstretched length
Limit of proportionalityPoint beyond which force is no longer proportional to extension
Elastic limitPoint beyond which the material no longer returns to its original length
Elastic potential energyEnergy stored in a stretched or compressed material, ½kx2

Avoid these mistakes

Using the total length instead of the extension — x is stretched length minus natural length.
Reading k as the gradient when the axes are swapped — then the gradient is 1/k.
Applying Hooke's law beyond the limit of proportionality — the graph curves there.
What the symbols mean:
Forgetting the ½ in elastic energy — E = ½kx2.
What the symbols mean: E = elastic potential energy stored (J) · k = spring constant (N m−1) · x = extension (m) · F = force (N)
Adding spring constants in series — add the reciprocals instead.

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