Solids, Springs and Hooke's Law
Forces on Solid Materials
To stretch, bend or compress a solid object, you need to apply more than one force. A single force acting on an object simply accelerates it — it takes at least two forces, acting in opposite directions, to change its shape. Think of pulling both ends of a rubber band: one hand pulls left, the other pulls right, and the band stretches between them.
The particle model explains why this happens. In a solid, particles are held in place by forces of attraction and repulsion between neighbouring particles. Stretching the material pulls particles further apart, increasing the attractive forces. Compressing it pushes particles closer together, increasing the repulsive forces. When the external force is removed, these inter-particle forces can restore the original shape — if the material has not been overstretched.
Stretching, bending or compressing a solid requires two forces acting in opposite directions. The inter-particle forces in the solid resist the change in particle separation, which is the microscopic origin of the restoring force.
Elastic and Plastic Deformation
Elastic deformation occurs when a material springs back to its original shape once the force is removed. The inter-particle forces are strong enough to restore the original arrangement. A stretched spring or a stretched rubber band can both show elastic deformation.
Plastic deformation occurs when the force is large enough to permanently rearrange the particles. Particles slip past each other into new positions, and when the force is removed the material stays in its new deformed shape. Bending a metal paperclip until it stays bent is an example of plastic deformation. The particle model distinguishes these two types: elastic deformation keeps inter-particle spacings within the restoring-force range, while plastic deformation moves particles to entirely new equilibrium positions.
Hooke’s Law
Robert Hooke, working in the seventeenth century, discovered that for many springs and elastic materials, the extension is directly proportional to the applied force — provided the material is not stretched too far. This is now called Hooke’s law.
The mathematical form of Hooke’s law is:
$$F = k \times e$$
Where \(F\) is the force applied in newtons (N), \(k\) is the spring constant in N/m, and \(e\) is the extension in metres (m). A stiffer spring has a higher spring constant: it requires a larger force to produce the same extension.
Hooke’s law: \(F = ke\). On a force–extension graph the relationship is a straight line through the origin up to the limit of proportionality. Beyond this point the graph curves, and if the spring is permanently deformed it has exceeded its elastic limit.
Hang a spring from a clamp stand and measure its natural (unstretched) length with a ruler. Add known masses one at a time and record the new length each time. Calculate the extension by subtracting the natural length from the stretched length.
Plot force (y-axis) against extension (x-axis). The straight-line section through the origin shows Hooke’s law is obeyed; the gradient of this section equals the spring constant \(k\). When the line begins to bend towards the extension axis, the limit of proportionality has been exceeded.
Worked example: A spring has a natural length of 15.0 cm. With a 200 g mass (weight ≈ 2.0 N) attached it stretches to 21.0 cm. Calculate the spring constant.
Extension: \(e = 21.0 - 15.0 = 6.0\) cm = 0.060 m. \(k = F \div e = 2.0 \div 0.060 \approx 33\) N/m.
Reduce parallax error when reading lengths by using a set square or a pointer at eye level. Always remove the loads gently after the experiment to check whether the spring has been permanently deformed.
Linear and Non-Linear Behaviour
Not all elastic materials show a linear (straight-line) force–extension relationship. A rubber band, for example, can stretch a great deal before the force rises sharply — its force–extension graph is a curve throughout, not a straight line at the start. Both the spring (linear up to its limit) and the rubber band (non-linear throughout) are elastic, because both return to their original shape, but only the spring obeys Hooke’s law in its initial range.
When work is done to stretch or compress a spring within its limit of proportionality, energy is stored as elastic potential energy. This can be recovered when the force is removed.
For a spring obeying Hooke’s law, the energy stored is:
$$E_e = \frac{1}{2} k e^2$$
Worked example: A spring with \(k = 50\) N/m is stretched by 0.10 m. Energy stored = \(\frac{1}{2} \times 50 \times 0.10^2 = \frac{1}{2} \times 50 \times 0.01 = 0.25\) J.
On a force–extension graph, the elastic potential energy equals the area under the straight-line section — a triangle with area \(\frac{1}{2} \times F \times e\), which is the same as \(\frac{1}{2} ke^2\) since \(F = ke\).
Elastic deformation: material returns to original shape; inter-particle forces restore the arrangement. Plastic deformation: particles slip permanently into new positions; the material does not recover. The boundary between them is the elastic limit of the material.