Hooke's law and the spring constant, stress, strain and the Young modulus, elastic strain energy, density and upthrust, and viscous drag with Stokes' law and terminal velocity.

What this dot point is asking

Edexcel wants you to apply Hooke's law and the spring constant, define stress, strain and the Young modulus and measure it, calculate elastic strain energy, use density and upthrust, and analyse viscous drag through Stokes' law and the idea of terminal velocity.

The answer

Hooke's law and the spring constant

The relationship is linear up to the limit of proportionality. Beyond the elastic limit a material is permanently deformed and does not return to its original length when the load is removed.

Stress, strain and the Young modulus

The Young modulus measures stiffness independent of the sample's shape. It is the gradient of the straight-line part of a stress-strain graph. The standard Edexcel core practical loads a long thin wire over a pulley, measures extension with a marker and ruler (or vernier), plots stress against strain, and takes the gradient. A long wire gives a measurable extension; a thin wire gives a high stress for modest loads.

Elastic strain energy

Density, upthrust and viscous drag

Density is $\rho = \frac{m}{V}$. An object immersed in a fluid feels an upthrust equal to the weight of fluid it displaces (Archimedes' principle), $U = \rho_f V g$. A small sphere moving slowly through a viscous fluid feels a retarding drag given by Stokes' law:

As a falling sphere speeds up, the drag grows until it balances the net downward force; the sphere then falls at a constant terminal velocity. Setting weight equal to upthrust plus drag and solving for $v$ gives $v_{\text{term}} = \frac{2r^2 g(\rho_s - \rho_f)}{9\eta}$.

Examples in context

Bridge and building engineers choose materials by their Young modulus and yield stress so structures flex little under load and never reach the elastic limit. A bungee cord stores elastic strain energy that decelerates the jumper. Ships float because their hull displaces enough water for the upthrust to equal their weight. The falling-ball method measures the viscosity of oils and syrups using terminal velocity, and Stokes' law also governs how slowly fine dust or fog droplets settle through air.

Try this

Q1. Define the Young modulus. [1 mark]

• Cue. The ratio of tensile stress to tensile strain in the linear (Hookean) region of the material.

Q2. A spring extends $5.0$ cm under a $10$ N load. Find the spring constant. [2 marks]

• Cue. $k = \frac{F}{x} = \frac{10}{0.050} = 200$ N per metre.

Q3. State the condition for an object falling through a fluid to be at terminal velocity. [2 marks]

• Cue. The net force is zero: the weight equals the upthrust plus the viscous drag, so the object falls at constant speed.