Foundations — Manometers, barometers

This page assumes nothing. Before you can read the parent note on manometers & barometers, you must own six little ideas: what pressure really is, what density is, what weight is, what a column looks like, what depth $h$ means, and why "same level = same pressure." We build each one from a picture.

1. Pressure $P$ — a push spread over an area

Why divide by area? Picture pressing a drawing pin. Your thumb pushes with the same force on the flat head and on the sharp tip — but the tip has a tiny area, so its pressure is huge and it pierces. Same push, different area, different result. Pressure is what your skin actually "feels."

2. Density $\rho$ — how much stuff is packed in

Picture two identical boxes: one filled with feathers, one with lead. Same box (same volume $V$), but the lead one is far heavier (bigger mass $m$). Lead has more mass crammed into the same space — higher density.

3. Weight $W$ — gravity's downward pull on mass

Careful: mass ($m$, in kg) is how much stuff there is; weight ($W$, in N) is how hard gravity tugs on that stuff. On the Moon your mass is unchanged but your weight shrinks because $g$ is smaller.

4. Volume of a column $V=Ah$ — turning a shape into a number

Now stack ideas 1–3. Imagine a straight tube of liquid: a column. Its cross-section (the circle you'd see looking down the tube) has area $A$; its height is $h$. The volume of that column is simply

$V=A\times h.$

Think of stacking coins: each coin has face area $A$, and a stack of height $h$ has volume "area × height." Liquid is the same — the tube is a stack of thin liquid discs.

Combine everything so far — the weight of that liquid column:

$W=mg=(\rho V)g=\rho (Ah)g=\rho Ahg.$

5. Depth $h$ — and why only the vertical part counts

Why vertical only? Gravity points straight down. A liquid disc's weight depends on how much liquid is stacked directly above it against gravity. If the tube leans over at an angle $\theta$, the liquid travels a longer slanted length $L$, but the vertical rise is only $h=L\sin \theta$.

6. "Same fluid, same level ⇒ same pressure"

This is the last building block, and it's the one that lets you read a U-tube. Inside one connected body of still liquid, any two points at the same height feel the same pressure.

Why must this be true? Suppose the two points at the same level had different pressures. Then the higher-pressure side would push liquid sideways toward the lower-pressure side — the liquid would flow. But we assumed the fluid is static (not moving). The only way to have no sideways flow is for the pressures at equal height to be equal. This idea rests on Pascal's Law.

Static equilibrium — the hidden assumption

Everything above secretly assumes the fluid is in static equilibrium: it is at rest, and every little parcel has zero net force on it (all the pushes balance). That's why we could say up-forces = down-forces when deriving $\Delta P=\rho gh$. The moment fluid moves, you need the bigger machinery of Bernoulli's Equation instead. For manometers and barometers, nothing flows — static is exactly right.

Prerequisite Map

Equipment checklist

Test yourself — you're ready for the parent note when you can answer each without peeking.

Connections

• Pascal's Law — justifies "same level ⇒ same pressure."
• Hydrostatic Pressure — the $\rho gh$ result these foundations assemble.
• Density and Specific Gravity — the $\rho$ that sets column height.
• Atmospheric Pressure — what the barometer ultimately reads.
• Buoyancy and Archimedes Principle — also built on depth-dependent pressure.
• Bernoulli's Equation — what replaces this when the fluid moves.