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3.7.1Forces and Motion

Deformation and Compression

5 min read 9 parts Lesson 23 of 66 1 quiz

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Lesson summary

3.7.1 Deformation and Compression — OCR A Level Physics A notes: Describe tensile and compressive forces, apply Hooke's law to springs and wires, and calculate the energy stored.Forces that deform a solid come in two basic kinds.

  • force constant
  • limit of proportionality
  • elastic limit
  • Worked example 1
  • Worked example 2
  • Worked example 3

Section 1 of 9Tensile and compressive forces

Forces that deform a solid come in two basic kinds.

Tensile
A tensile force pulls on an object and tries to stretch it (a cable holding a lift).
Compressive
A compressive force pushes inward and tries to squash it (the pillars of a bridge).
Both change the separation of the atoms or molecules: under tension the average spacing increases and the attractive bonds pull back; under compression the spacing decreases and repulsion between atoms pushes back.
The restoring force from the material is the reason a deformed solid pushes back on whatever deforms it.

A single object can experience both at once: when a beam bends, its top surface is in compression and its bottom surface in tension, while the middle layer is hardly stretched at all.

Section 2 of 9Hooke's law

For many springs and wires the extension x is proportional to the applied force F, provided a limit is not exceeded: F = kx, where k is the force constant (spring constant) in N m−1, F is the force in N and x the extension in m.

Hooke's law
The extension is the increase in length over the natural (unstretched) length, not the total length.
A stiff spring has a large k.
The same law applies to compression, with x then the decrease in length.

The graph of force against extension is a straight line through the origin while Hooke's law holds, and its gradient is k.

Limit of proportionality
The point beyond which the line stops being straight is the limit of proportionality.
Elastic limit
Just after that, there is typically an elastic limit, beyond which the object does not return to its original length when the force is removed (see 3.7.8).

Section 3 of 9Springs in combination

Springs in series (end to end) carry the same force and their extensions add: 1/k = 1/k1 + 1/k2 + …, so the combination is softer.

Parallel
Springs in parallel share the force and have the same extension: k = k1 + k2 + …, so the combination is stiffer.
These rules mirror the equations for resistors, with k analogous to conductance in parallel.
For example, two identical springs of constant 40 N m−1 in series have k = 20 N m−1, and in parallel 80 N m−1.

Section 4 of 9Energy stored

The work done in stretching a spring equals the area under the force–extension graph.

1
While Hooke's law applies this is a triangle, so the energy stored is E = ½Fx = ½kx2, in joules.
2
If the force is removed from an elastic spring this energy can be recovered as kinetic energy, as in a catapult or a toy dart gun.
3
If the graph is not linear, count squares or use the trapezium rule to find the area.
4
For an object that is permanently deformed, the unloading line differs from the loading line; the area between them is energy dissipated as thermal energy.

Doubling the extension of a spring multiplies the stored energy by four, because E ∝ x2, while the force only doubles.

Section 5 of 9Investigating a spring

Hang a spring vertically, record its natural length with a rule next to a pointer or a marker fixed at the bottom, then add 100 g masses one at a time.

1
Weight F = mg.
2
Record the length each time and calculate extension.
3
Plot F against x, draw a line of best fit, and find k from the gradient.
4
Take care to read the rule at eye level to avoid parallax, avoid overloading the spring, and unload to check for permanent set.
5
For a rubber band the graph is a curve that loops, showing that it is not Hookean.

Section 6 of 9Links to other topics

The ideas here lead directly to stress, strain and the Young modulus in the next lessons: the force constant depends on the dimensions of a sample, while the Young modulus does not.

1
Hooke's law applies only up to the limit of proportionality, and elastic behaviour continues a little further.
2
Springs also appear in oscillation and energy contexts: a mass on a spring exchanges elastic and kinetic energy.
3
When using F = kx in problems with weights, remember F = mg and that the extension, not the total length, is required.
4
The energy stored is shown by the area under the graph even for a non-linear curve.

Section 7 of 9Worked examples

  1. A spring of constant 250 N m−1 is stretched by 0.060 m. Find the force and the stored energy.

  2. A 0.50 kg mass hung from a spring extends it from 0.20 m to 0.26 m. Find k and then the extension if a 1.5 kg mass is used.

  3. Two springs of k = 60 N m−1 and 30 N m−1 are connected in series and a force of 12 N applied. Find the total extension.

Section 8 of 9Key terms

TermMeaning
Tensile forceA pulling force that tends to stretch an object.
Compressive forceA pushing force that tends to squash an object.
Force constantForce per unit extension, k = F/x, in N m⁻¹.
Limit of proportionalityPoint beyond which force is no longer proportional to extension.
ExtensionIncrease in length from the natural length.
Elastic potential energyEnergy stored in a deformed elastic object, ½kx².

Section 9 of 9Exam technique

1
Using the total length as the extension in F = kx.
2
Believing a larger force always gives a proportionally larger extension; this fails beyond the limit of proportionality.
3
Forgetting to square the extension in E = ½kx².
4
Adding spring constants for series springs; that is for springs in parallel.
5
Reading the gradient of an extension–force graph as k; there the gradient is 1/k.
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