Manometers, barometers

WHY do these devices work at all?

Deriving the hydrostatic pressure law (from scratch)

Take a fluid cylinder: cross-sectional area $A$ , top at depth $h_1$ , bottom at depth $h_2$ ( $h_2>h_1$ ). Forces along the vertical:

Pressure on top pushes down: $P_1 A$
Pressure on bottom pushes up: $P_2 A$
Weight pulls down: $W = mg = \rho \, V \, g = \rho A (h_2-h_1) g$

Why this step? In equilibrium the net force is zero, so up-forces = down-forces.

$P_2 A = P_1 A + \rho A (h_2-h_1) g$

Divide by $A$ (it cancels — why? because pressure is force per area, so area never matters for the relationship):

$\boxed{P_2 = P_1 + \rho g (h_2 - h_1)}$

The Barometer — measuring atmospheric pressure

Derivation. Pick two points at the same height: point A inside the tube at the dish surface, point B in the open dish surface.

At B (open surface): pressure $= P_{atm}$ .
At A (same level, inside tube): pressure from the column above $=$ (vacuum pressure at top) $+ \rho g h = 0 + \rho g h$ .

Why equal? Same fluid, same horizontal level ⇒ same pressure (otherwise fluid would flow sideways).

$P_{atm} = \rho_{Hg}\, g\, h$

The Manometer — measuring gauge pressure

Derivation. Choose the lowest common level of liquid — call it the reference line. Pressure at this line is the same on both arms (connected, same fluid, same height).

Left arm (gas side), going down from gas to line: $P_{left} = P_{gas} + \rho g h_{gas\,col}$
Right arm (open side): $P_{right} = P_{atm} + \rho g h_{open\,col}$

Setting $P_{left}=P_{right}$ and simplifying so $h$ is the difference in the two free surfaces:

$\boxed{P_{gas} = P_{atm} + \rho g h}$

where $h$ is positive if the open arm is higher (gas pushes its side down).

Steel-manned mistakes

80/20 — the 20% that gives 80%

Recall Feynman: explain to a 12-year-old

Imagine diving into a pool. Near the surface your ears feel fine, but go deep and they hurt — that "hurt" is pressure, and it grows because more water is sitting on top of you. Now flip it around: if we know how much your ears hurt, we can guess how deep you are! A barometer and manometer do exactly this with a thin tube of liquid. Air (or trapped gas) pushes on the liquid and makes it rise; we just measure how tall the liquid climbs, and that height is the pressure. Mercury is used because it's super heavy, so even strong air pressure only lifts it a little — easy to fit in a tube on your desk.

Flashcards

Hydrostatic pressure formula with depth

h

\Delta P=\rho g h

(only vertical height counts).

Why does tube cross-sectional area not affect pressure?

Pressure = force/area; extra fluid weight sits on proportionally more area, so

A

cancels.

What sits at the top of a mercury barometer tube?

A vacuum (

P\approx0

), which is why it reads absolute

P_{atm}=\rho g h

Standard atmospheric pressure in mmHg and Pa?

760 mmHg

\approx 1.013\times10^5

Pa.

Why mercury instead of water in a barometer?

Mercury is 13.6× denser, so column height (

h=P/\rho g

) is ~0.76 m vs ~10.3 m for water.

Open manometer: gas pressure formula?

P_{gas}=P_{atm}\pm\rho g h

;

+

if open arm higher,

-

if gas arm higher.

Does an open-tube manometer read gauge or absolute pressure?

Gauge (

\rho g h

); add

P_{atm}

for absolute.

For a tube inclined at angle

\theta

with liquid length

L

, what is

h

h=L\sin\theta

(vertical component only).

Two points in the same connected static fluid at the same height have...?

The same pressure.

Definition of gauge pressure?

P_{gauge}=P_{abs}-P_{atm}=\rho g h

Connections

Pascal's Law — pressure transmits equally; basis of "same level = same pressure".
Hydrostatic Pressure — the parent $\rho g h$ result.
Buoyancy and Archimedes Principle — also rests on depth-dependent pressure.
Atmospheric Pressure — what the barometer measures.
Bernoulli's Equation — generalizes pressure to moving fluids.
Density and Specific Gravity — choice of manometric fluid ( $\rho$ ).

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, manometer aur barometer dono ek hi simple cheez par chalte hain: fluid ka column ek pressure-measuring stick hai. Jitna neeche jaoge, upar utna zyada fluid ka weight tumhare upar baithta hai, isliye pressure badhta hai. Yeh badhne ka rate hota hai $\Delta P = \rho g h$ — yaha sirf vertical height $h$ matter karti hai, tube ki motai (area) ya shape se koi farak nahi padta, kyunki pressure to force/area hota hai aur area cancel ho jaata hai.

Barometer mein ek tube ko mercury se bharke ulta dish mein dubo dete hain. Upar vacuum ( $P=0$ ) hota hai, aur neeche atmosphere mercury ko upar push karta hai. Balance condition: $P_{atm}=\rho_{Hg}\,g\,h$ . Mercury isliye use karte hain kyunki woh bahut dense hai — paani ka column to 10 meter lamba ho jaata, mercury ka sirf 0.76 m (760 mmHg = 1 atm). Yaad rakho: barometer absolute pressure deta hai kyunki top par vacuum hota hai.

Manometer ek U-tube hai. Ek arm gas/vessel se juda, doosra arm atmosphere ke liye open. Dono liquid surfaces ki height difference $h$ se pressure milta hai: $P_{gas}=P_{atm}\pm\rho g h$ . Agar open arm uncha hai to gas pressure zyada hai (plus sign), agar gas arm uncha hai to gas pressure atmosphere se kam (minus). Yeh device direct gauge pressure ( $\rho g h$ ) deta hai; absolute chahiye to $P_{atm}$ add kar do.

Trick yaad rakhne ka: "same fluid, same level, same pressure". Hamesha ek common reference line choose karo jaha dono arms connected ho, aur dono taraf se pressure equal likho — sab questions isi se ban jaate hain. Inclined tube mein dhyaan: $h = L\sin\theta$ , slanted length nahi, sirf vertical rise.

WHY do these devices work at all?

Deriving the hydrostatic pressure law (from scratch)

The Barometer — measuring atmospheric pressure

The Manometer — measuring gauge pressure

Steel-manned mistakes

80/20 — the 20% that gives 80%

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Fluid Mechanics

Connections