Forces, density and pressure

Density is a fundamental property of matter that tells us how much 'stuff' (mass) is packed into a given space (volume). A small, heavy object is very dense, while a large, light object is not. For example, a kilogram of lead takes up much less space than a kilogram of feathers because lead is much denser. Density is an intrinsic property of a substance, meaning a pure substance will have the same density regardless of how much of it you have. To find the density (ρ) of an object, you measure its mass(m) and its volume (V) and use the formula ρ = m/V. The standard unit for density is kilograms per cubic metre (kg/m³), although grams per cubic centimetre (g/cm³) is also common.

Pressure is defined as the force acting perpendicularly on a unit area of a surface. A large force on a small area produces a high pressure, which explains why a sharp knife (small area) cuts easily, while a blunt knife (larger area) does not. The formula is p = F/A, where 'p' is pressure, 'F' is the normal force, and 'A' is the cross-sectional area over which the force acts. The term 'normal' is crucial – it means the force is at a right angle (90°) to the surface. The SI unit of pressure is the Pascal (Pa), which is defined as one newton per square metre (N/m²).

In a fluid (a liquid or a gas), pressure increases with depth. This is because the fluid at a lower level has to support the weight of all the fluid above it. The pressure difference (Δp) between two points in a fluid is determined by the fluid's density (ρ), the acceleration of free fall (g), and the vertical height difference (Δh) between the points. This relationship is given by the equation Δp = ρgΔh. This pressure is often called 'gauge pressure' as it is the pressure relative to the surroundings. To find the absolute pressure at a certain depth, you must add the atmospheric pressure to the gauge pressure.

When an object is submerged in a fluid, it experiences an upward force called upthrust. This force arises because the pressure on the bottom surface of the object is greater than the pressure on its top surface (since pressure increases with depth). Archimedes' principle provides a simple way to calculate this force: it states that the upthrust on a submerged object is equal to the weight of the fluid that the object displaces. If the upthrust is greater than the object's weight, the object will float or rise. If the upthrust is less than the object's weight, it will sink. If they are equal, the object will float in equilibrium.

When forces are applied to a solid object, they can cause it to change shape, a process called deformation. A tensile force stretches an object, while a compressive force squashes it. For many materials, like springs and wires, the extension(x) is directly proportional to the applied force (F), provided a certain limit is not exceeded. This relationship is Hooke's Law, F = kx. The constant 'k' is the force constant (or spring constant), a measure of the object's stiffness. A graph of force against extension is a straight line through the origin up to the 'limit of proportionality'. If the force is removed within the 'elastic limit', the object returns to its original shape (elastic deformation). Beyond this limit, the object is permanently deformed (plastic deformation).

To compare the 'stretchiness' of materials without being confused by their shape or size, we use stress and strain. Stress (σ) is a measure of the force applied per unit of cross-sectional area (σ = F/A), essentially normalising the force. Strain (ε) is the fractional change in length (ε = x/L), where x is the extension and L is the original length. Strain is a dimensionless quantity. For an elastic material, the ratio of stress to strain is a constant known as the Young Modulus (E). E = σ/ε. The Young Modulus is a measure of a material's stiffness; a higher value means the material is stiffer and harder to stretch. To measure E for a wire, one applies known forces (weights), measures the extension with a precise instrument (like a vernier scale), and measures the wire's original length and diameter (to find area).

When you stretch or compress an elastic material, you do work on it. This work is stored in the material as elastic potential energy (also called strain energy). When the force is removed, this stored energy is released, often causing the object to spring back. The amount of work done (and energy stored) is equal to the area under the force-extension graph. For a material that obeys Hooke's Law, the graph is a triangle. The area of a triangle is ½ × base × height, which for the graph corresponds to ½ × extension × force. This gives the formula for elastic potential energy, E_p = ½Fx. Since F = kx, we can substitute this in to get an alternative formula, E_p = ½kx².