Materials Under Stress and Density
Stretching and Deformation
When forces are applied to an object, they can change its shape. To stretch, bend or compress an object, at least two forces are needed — a single force would simply accelerate the object. When two opposing forces act on a spring (for example, a weight pulling it down while a support holds it up), the spring deforms. There are two types of deformation:
• Elastic deformation — the material returns to its original shape and size when the force is removed. The spring 'bounces back'.
• Plastic deformation — the material is permanently deformed. It has been stretched beyond its elastic limit and will not return to its original shape even when the force is removed.
Elastic deformation is reversible — the object returns to its original shape. Plastic deformation is permanent — the object remains deformed after the force is removed.
Hooke's Law
Hooke's Law states that the extension of a spring is directly proportional to the force applied, as long as the elastic limit is not exceeded. The equation is:
$$F = k \times e$$
Where \(F\) is force in newtons (N), \(k\) is the spring constant (N/m), and \(e\) is the extension in metres (m). Extension is the change in length — \(e = \text{stretched length} - \text{original length}\). A higher spring constant means a stiffer spring that is harder to stretch. The equation can be rearranged to find spring constant (\(k = F/e\)) or extension (\(e = F/k\)).
On a force–extension graph, Hooke's Law gives a straight line through the origin. The gradient of this straight line equals the spring constant \(k\). Beyond the elastic limit, the graph curves — the extension increases more rapidly for the same increase in force, and the spring undergoes plastic deformation.
Question: A spring has a spring constant of 250 N/m. A force of 15 N is applied. Calculate the extension, assuming the elastic limit is not exceeded.
Step 1: Rearrange: \(e = \frac{F}{k} = \frac{15}{250} = 0.06 \text{ m}\) (6 cm)
Check units: F in newtons, k in N/m, e in metres.
Now try: A spring extends by 0.04 m when a force of 20 N is applied. What is the spring constant? (Answer: \(k = 20/0.04 = 500 \text{ N/m}\))
A standard investigation hangs a spring from a fixed support and adds weights one at a time, recording the extension after each addition. Measure the original length of the spring before adding any load, then measure the length after each mass is added. Extension = current length − original length. Plot force (y-axis) against extension (x-axis) — a straight line through the origin confirms Hooke's Law is obeyed. The gradient gives the spring constant.
Safety note: do not overload the spring — once plastic deformation occurs, the spring cannot return to its original length and results will be unreliable. Use a clamp stand secured to the bench, and place a cushion or tray below the masses in case the spring snaps.
Density
Density is the mass per unit volume of a material. It tells you how much matter is packed into a given space. The equation is:
$$\rho = \frac{m}{V}$$
Where \(\rho\) (Greek letter rho) is density in kg/m³, \(m\) is mass in kg and \(V\) is volume in m³. In everyday calculations, g/cm³ is also used. To convert: 1 g/cm³ = 1 000 kg/m³.
Density explains why objects float or sink. An object floats in a fluid if its average density is less than the fluid's density. Ice (density approximately 0.92 g/cm³) floats on water (1.00 g/cm³) because it is less dense. Steel ships float because their overall average density — including the air inside the hull — is less than water.
The particle model explains density: denser materials have more mass packed into the same volume — either because their atoms are heavier, or because the atoms are packed more tightly. Solids are generally denser than liquids (they are more tightly packed), and liquids are denser than gases. Water is an exception — ice is less dense than liquid water because ice has a more open crystal lattice structure.
For a regular solid (cuboid, cylinder), calculate volume from dimensions and measure mass using a balance. For an irregular solid, use water displacement: partially fill a measuring cylinder with water and record the volume. Submerge the solid and record the new volume. The volume of the solid = new volume − original volume. Mass ÷ volume = density.
For a liquid, measure a known volume using a measuring cylinder, then measure the mass of the liquid alone (subtract the cylinder mass). Density = mass ÷ volume.
Hooke's Law: \(F = ke\). Spring constant = gradient of force–extension graph. Density: \(\rho = m/V\). Objects float when their average density is less than the surrounding fluid's density.