Physics · Lesson 9

Elasticity & Hooke's law

JAMBWAECNECO

What the syllabus asks

JAMB: Elasticity: elastic limit, yield point, breaking point, Hooke’s law and Young’s modulus, springs balance as a device for measuring force, work done in springs and elastic strings

WAEC: Elastic properties of solids: elasticity, Hooke’s law, elastic limit, yield point, breaking point, force–extension graphs, spring constant, stress, strain and Young’s modulus, work done (energy stored) in a stretched spring or wire, springs in series and parallel, applications

NECO: Elastic properties of solids: elasticity, Hooke’s law, elastic limit, yield point, breaking point, force–extension graphs, spring constant, stress, strain and Young’s modulus, work done (energy stored) in a stretched spring or wire, springs in series and parallel, applications

What you will learn

Distinguish between elastic and plastic behaviour and give everyday examples.
State Hooke’s law and use F = ke to solve problems.
Describe an experiment to verify Hooke’s law and find a spring constant from a graph.
Identify the proportional limit, elastic limit, yield point and breaking point on a force–extension graph.
Define stress, strain and Young’s modulus and calculate each.
Calculate the energy stored in a stretched spring or wire.
Find the effective spring constant of springs in series and in parallel, and explain applications such as spring balances and vehicle suspensions.

Key points

Elastic: returns to original shape; plastic: permanent deformation.
Hooke’s law: F = ke up to the proportional limit; e is the extension, not the length.
k = gradient of a force–extension graph; unit N/m.
Key points on the graph: proportional limit, elastic limit, yield point, breaking point.
Ductile materials have a large plastic region; brittle materials break with little plastic stretch.
Stress = F/A (Pa); strain = e/L (no unit).
Young’s modulus E = stress/strain = FL/(Ae); it depends only on the material.
Energy stored = ½Fe = ½ke² = area under the F–e graph; per unit volume = ½ × stress × strain.
Series springs: 1/k = 1/k₁ + 1/k₂; parallel springs: k = k₁ + k₂.
Spring balances have uniform scales because e ∝ F.

Start here: Elastic and plastic behaviour

Stretch a rubber band around a bundle of ₦500 notes and it springs back when you take it off. Press a lump of plasticine or bend a paper clip far enough, and it stays bent. These everyday observations show two different ways materials respond to forces.

A force can change the shape or size of a body. If the body returns completely to its original shape and size when the force is removed, it is behaving elastically. If it keeps some permanent change, it has been plastically deformed.

Elastic examples: a spring in a mattress, a catapult rubber, a trampoline, the steel cables of the Third Mainland Bridge under traffic, a football when kicked.
Plastic examples: clay used by potters in Ilorin, chewing gum, putty, a car body panel dented in an accident, copper drawn into wire.

Most solids are elastic only up to a point. A steel spring is elastic for small loads, but if it is overloaded it becomes permanently stretched. The behaviour depends on the material and on how big the force is.

Why are solids elastic? The particles in a solid are held in place by attractive and repulsive forces between neighbouring particles. Pulling the solid moves the particles slightly apart, and the attractive forces pull them back when the force is removed, just like tiny springs joining the particles. If the force is too large, layers of particles slide past one another into new positions and cannot return: that is plastic deformation.

Worked example 9.1

Classify each as mainly elastic or plastic: (a) a bow string, (b) wet clay, (c) a diving board, (d) a nail bent by a hammer.

(a) Elastic: it returns after the arrow is released.
(b) Plastic: it keeps the shape it is pressed into.
(c) Elastic: it springs back after the diver leaves.
(d) Plastic: the bend is permanent, because the force exceeded the elastic limit of the steel.

(a) elastic, (b) plastic, (c) elastic, (d) plastic

⚠️ Common mistake: thinking “elastic” means “stretches a lot”. Steel stretches very little but is highly elastic within its limit; plasticine stretches easily but is not elastic at all.

Try it yourself. Why does a stretched spring that has been overloaded stay longer than before?

Show solution

The load exceeded its elastic limit, so layers of particles slipped into new positions and the deformation became plastic (permanent).

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Practice questions

1. A force of 8 N stretches a spring by 4 cm. Find the spring constant.

Show worked answer

k = 8/0.04 = 200 N/m.

2. A spring of constant 150 N/m carries a 300 g mass. Find the extension. (g = 10 m/s²)

Show worked answer

3 N ÷ 150 = 0.02 m = 2 cm.

3. A spring of natural length 25 cm is 31 cm long under a 3 N load. Find its length under 5 N.

Show worked answer

k = 3/0.06 = 50 N/m; e = 5/50 = 0.10 m; length = 35 cm.

4. Find the energy stored in a spring of constant 400 N/m stretched by 5 cm.

Show worked answer

½ × 400 × 0.05² = 0.5 J.

5. A load of 400 N hangs from a wire of cross-sectional area 2 × 10⁻⁶ m². Find the stress.

Show worked answer

2 × 10⁸ Pa.

Key words

Elasticity: The ability of a material to regain its original shape and size after the deforming force is removed.
Plastic deformation: A permanent change of shape that remains after the force is removed.
Extension: The increase in length produced by a stretching force.
Hooke’s law: Extension is directly proportional to the applied force, provided the proportional limit is not exceeded.
Spring constant (k): The force needed per unit extension; unit N/m. Also called force constant or stiffness.
Proportional limit: The greatest force for which extension is still proportional to force.
Elastic limit: The greatest force a material can take and still return to its original length.
Yield point: The point beyond the elastic limit where the material suddenly stretches a lot for little extra force.
Breaking point: The point at which the material snaps.
Tensile stress: Force per unit cross-sectional area, F/A; unit N/m² (pascal).
Tensile strain: Extension per unit original length, e/L; no unit.
Young’s modulus: The ratio of tensile stress to tensile strain for a material, within the proportional limit.

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