The Timpson Lyceum · University Physics I (2025)

Fluid Mechanics: Statics and Dynamics – Lecture 12 (University Physics I)

Overview

This lecture introduces fluid mechanics, covering both fluid statics (fluids at rest) and fluid dynamics (fluids in motion). A fluid is anything that flows — for this course, that means liquids in containers and the atmosphere. We build up the core static results: density, pressure, how pressure grows with depth, Pascal's principle (and the hydraulic lift), the difference between gauge and absolute pressure, and Archimedes' principle of buoyancy. We then move to flowing fluids, deriving the continuity equation and Bernoulli's equation, the single relation from which most of the chapter's results follow. These ideas explain everyday puzzles like why a straw works, why ships made of steel float, and the counterintuitive fact that faster-moving fluid has lower pressure.

Key Concepts

Density

• Density $\rho$ is mass per unit volume, measured in $\text{kg/m}^3$: $\rho = \dfrac{m}{V}$, so $m = \rho V$.
• Reference values: air $\approx 1.2\ \text{kg/m}^3$, water $= 1000\ \text{kg/m}^3$, concrete $\approx 2000\ \text{kg/m}^3$.
• For our purposes liquids are incompressible — water's density is essentially the same at the surface and deep underwater. Increasing depth raises pressure, not density.

Pressure

• Pressure is force per unit area acting on a surface: $P = \dfrac{F}{A}$, measured in pascals ($1\ \text{Pa} = 1\ \text{N/m}^2$).
• Pressure is a scalar, and the force it produces always acts perpendicular to the surface it contacts.
• One atmosphere $\approx 1.013\times10^{5}\ \text{Pa} \approx 14.7\ \text{psi}$. The weight of the atmosphere above you is comparable to about 10 m of water — which is why a straw works (the outside atmosphere pushes liquid up the low-pressure region you create), and why the longest practical straw is about 10 m.

Pressure and depth

• Analyzing a thin slice of fluid in equilibrium gives the differential relation $\frac{dP}{dy} = -\rho g$
• Integrating between two points (with $\rho$ and $g$ constant) yields the working formula for the pressure a depth $h$ below a surface: $P = P_0 + \rho g h$
• Key consequence: pressure depends only on depth, not on the shape of the container or the total volume of fluid. The same depth gives the same pressure whether you are in a vast lake or a thin tube (this is why a dam must withstand the same pressure regardless of how large the reservoir behind it is).

Pascal's principle

• Pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the container walls.
• Hydraulic lift: equal pressure on both pistons means $\dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}$, so $F_2 = \dfrac{A_2}{A_1}F_1$. A small force on a small piston produces a large force on a large piston. Energy is conserved: since the displaced volumes are equal ($A_1 d_1 = A_2 d_2$), the large force moves a proportionally smaller distance, so $F_1 d_1 = F_2 d_2$.

Gauge vs. absolute pressure

• A gauge reads the difference between the fluid's pressure and the atmosphere. A flat tire reads 0 psi even though its absolute pressure is about 1 atm.
• Absolute pressure = gauge pressure + 1 atm. Use $P_0 = P_{\text{atm}}$ in $P = P_0 + \rho g h$ for absolute pressure, or $P_0 = 0$ for gauge pressure.

Buoyancy and Archimedes' principle

• The buoyant force is the net upward force from the fluid pressure being larger on the bottom of a submerged region than on top. It equals the weight of the fluid displaced: $F_B = m_{\text{fluid}}\, g = \rho_{\text{fluid}}\, V_{\text{displaced}}\, g$
• An object floats if its average density is less than the fluid's, and sinks if greater. A steel hull floats because the air it encloses makes its average density less than water, even though steel itself is denser.

Continuity equation (mass conservation)

• For an incompressible fluid in steady flow, the same mass passes every cross-section per unit time, giving $A_1 v_1 = A_2 v_2$
• Cross-sectional area and flow speed are inversely related: a narrower pipe means faster flow (like covering part of a hose nozzle). The volume flow rate is $\dfrac{dV}{dt} = A v$.

Bernoulli's equation (energy conservation)

• Applying the work–energy theorem to a flowing ideal fluid (no viscosity, steady flow) gives the conserved quantity $P + \tfrac{1}{2}\rho v^2 + \rho g y = \text{constant}$
• The three terms are pressure, kinetic energy per unit volume, and gravitational potential energy per unit volume. Most chapter results are special cases:
  • Setting all velocities to zero recovers the hydrostatic relation $P_1 + \rho g y_1 = P_2 + \rho g y_2$.
  • At constant height, higher speed means lower pressure — counterintuitive but central. This explains the floating ping-pong ball in an airstream, the curve of a spinning ball, and part of the lift on a wing (faster air over the top → lower pressure → upward push).
• Torricelli's result: for an open tank draining through a small hole a depth $h$ below the surface, $v = \sqrt{2gh}$ — the same speed an object would reach falling from height $h$.

Worked Example

Tension in a cable holding a submerged gold statue. A 15 kg gold statue ($\rho_{\text{gold}} = 19.3\times10^{3}\ \text{kg/m}^3$) hangs from a cable, fully immersed in salt water ($\rho_{\text{water}} = 1.03\times10^{3}\ \text{kg/m}^3$). Find the cable tension.

In equilibrium the upward tension and buoyant force balance the weight: $T + F_B = mg$. Solving for tension and substituting $F_B = \rho_{\text{water}}\,V\,g$ with the statue's volume $V = m/\rho_{\text{gold}}$:

The buoyant force removes only about 8 N, so it is slightly easier to hold the dense gold statue underwater than in air — but not by much, because gold is far denser than water.

Summary

• A fluid is anything that flows; the key static quantities are density $\rho = m/V$ and pressure $P = F/A$ (a scalar acting perpendicular to surfaces, measured in pascals).
• Pressure increases with depth as $P = P_0 + \rho g h$ and depends only on depth, not container shape or fluid volume.
• Pascal's principle transmits applied pressure undiminished throughout an enclosed fluid, enabling the hydraulic lift to multiply force while conserving energy.
• Archimedes' principle: the buoyant force equals the weight of displaced fluid, $F_B = \rho_{\text{fluid}} V g$; objects float or sink according to their average density relative to the fluid.
• For flowing incompressible fluids, the continuity equation $A_1 v_1 = A_2 v_2$ conserves mass — narrower pipes flow faster.
• Bernoulli's equation $P + \tfrac{1}{2}\rho v^2 + \rho g y = \text{constant}$ conserves energy, implying faster flow means lower pressure and yielding results like the draining-tank speed $v = \sqrt{2gh}$.