Revise Density and pressure for Cambridge 9702 (4.3)

Density is a measure of how much 'stuff' (mass) is squeezed into a certain amount of space (volume). Imagine two boxes of the same size; one filled with feathers and the other with rocks. The box of rocks has a much higher density because there's more mass in the same volume. To measure the density of a regular solid, you can measure its dimensions to calculate volume and use a balance for its mass. For an irregular solid, you can find its volume by submerging it in a measuring cylinder with water and observing the volume of water displaced.

$\rho = \frac{m}{V}$

Pressure describes how a force is spread out over an area. If you push on a wall with your flat hand, the force is distributed over a large area, resulting in low pressure. If you push with just your fingertip, the same force concentrated on a tiny area creates much higher pressure. It's important to note that while force is a vector, pressure is a scalar quantity; it has magnitude but no direction. At any point within a fluid, pressure is exerted equally in all directions.

$P = \frac{F}{A}$

When you dive into a swimming pool, you feel the pressure increase as you go deeper. This is because the fluid above you is exerting a weight. The deeper you go, the more fluid is above you, and thus, the greater the pressure. This hydrostatic pressure can be derived by considering a vertical column of fluid of height $\Delta h$ and cross-sectional area $A$. The volume of this column is $V = A \Delta h$, and its mass is $m = \rho V = \rho A \Delta h$. The weight of the column is $W = mg = \rho A \Delta h g$. The pressure exerted by this weight is $p = \frac{W}{A} = \frac{\rho A \Delta h g}{A} = \rho g \Delta h$. The total pressure at a depth is this gauge pressure plus the atmospheric pressure at the surface.

$\Delta p = \rho g \Delta h$

Have you ever tried to push a ball underwater? It feels like something is pushing it back up - that's upthrust! This buoyant force arises because the pressure at the bottom of a submerged object is greater than at its top, creating an overall upward push. According to Archimedes' principle, this upthrust is equal to the weight of the fluid the object displaces. An object's fate in a fluid is determined by the balance between its weight and the upthrust. If the object's density is less than the fluid's density, it will float. If its density is greater, it will sink. If the densities are equal, it will remain suspended at any depth it's placed.

$F_B = \rho_{fluid} g V_{displaced}$