Q.What is the pressure on a swimmer 10m below the surface of a lake? (g = 10 m/sec^2)
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Hydrostatic Pressure Balance
Hydrostatic Pressure Balance
Imagine you're standing at the bottom of a swimming pool. You feel pressure on your ears — and the deeper you go, the more intense that pressure becomes. Now think about a column of water above you: every kilogram of that water is being pulled down by gravity. That weight has to be supported by the water below it. The deeper you go, the more water is stacked above you, so the greater the weight pressing down.
That's the core intuition: pressure in a fluid at rest increases with depth because the fluid above has to be supported by the fluid below.
The Precise Statement
Hydrostatic pressure balance is the condition that holds for any fluid at rest in a uniform gravitational field. It says:
Pbelow=Pabove+ρgh
where:
- Pbelow is the pressure at a lower point,
- Pabove is the pressure at a higher point,
- ρ is the density of the fluid (assumed constant),
- g is the acceleration due to gravity,
- h is the vertical depth between the two points.
Equivalently, the pressure gradient in the vertical direction is:
dzdP=−ρg
where z increases upward. The minus sign tells you pressure decreases as you go up.
Why This Makes Sense
Take a thin horizontal slab of fluid of area A, thickness dz, at some depth. Its weight is dW=ρgAdz. For the slab to be in equilibrium (not accelerating), the net upward force from pressure must exactly balance this weight.
The upward force on the slab's bottom face is P(z)A, and the downward force on its top face is P(z+dz)A. The net upward force is:
P(z)A−P(z+dz)A=−dzdPAdz
Setting this equal to the weight ρgAdz gives:
−dzdP=ρg
which is exactly the differential form above.
This balance assumes the fluid is static — no flow, no acceleration. If the fluid moves, additional terms (like viscous forces or inertial effects) appear.
Key Implications
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Pressure depends only on depth, not on the shape of the container. A tall thin tube and a wide shallow tank give the same pressure at the same depth — because only the vertical height of fluid above matters.
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Pressure is the same at all points on the same horizontal level. If you move sideways at constant depth, ρgh doesn't change, so P doesn't change.
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Gases are compressible, so ρ is not constant. For air, the density changes with pressure itself, leading to an exponential decrease — but the same principle applies locally.
A Common Mistake …
The pressure at any depth inside a liquid depends on the weight of the fluid column above that point, added on top of the atmospheric pressure already pressing on the surface.
…
Use P = P0 + ρgh with ρ(water) = 1000 kg/m^3, g = 10 m/s^2, h = 10 m.
The pressure at a depth h below the free surface of a liquid is given by:
P = P0 + ρgh
where P0 is atmospheric pressure at the surface, ρ is the density of the liquid, and g is the acceleration due to gravity.
Given: h = 10 m, g = 10 m/s^2, ρ(water) = 1000 kg/m^3.
Extra (gauge) pressure due to the water column:
ρgh = 1000 × 10 × 10 = 1 × 10^5 Pa
Taking atmospheric pressure as P0 ≈ 1.013 × 10^5 Pa (standard value): …
- CBSE 2026Set ANNUAL1 markQ.Fill in the blank: The S.I. unit of pressure is ________.
›Reveal solutionSolution
The SI unit of pressure is the pascal (Pa) = 1 N/m².
Pressure P is defined as the force F acting normally (perpendicularly) per unit area A: P = F/A. Since force is measured in newtons and area in square metres, pressure has SI unit N/m², which is given the spe …
- CBSE 2026Set ANNUAL1 markMCQQ.Assertion (A) : Blood pressure in Humans is greater at the feet than at the Brain. Reason (R) : Pressure exerted by a Liquid depends on the height of Liquid column.(a) Both A and R are true and R is correct explanation of A.(b) Both A and R are true and R is not the correct explanation for A.(c) A is true but R is false.(d) A is false but R is true.
›Reveal solutionSolution
Blood pressure is greater at the feet because of the greater height of blood column above them, exactly as predicted by the hydrostatic pressure formula P = hρg — so R correctly explains A.
Checking Assertion (A): In a standing human, blood pressure measured at the feet is indeed greater than that measured near the brain/head. This is TRUE.
Checking Reason (R): The pressure at a depth h inside a liquid column of density ρ is given by:
P = P0 + hρg
where P0 is the pressure at the top of the column. This shows pressure increases with the height (depth) of the liquid column above the point of measurement. This statement is TRUE.
…
- CBSE 2026Set ANNUAL1 markMCQQ.Pressure on a swimmer at a depth of 10 m below the surface of a lake will be (density of water = 1000 kg/m^3, g = 10 m/s^2)(a) 1 atm(b) 2 atm(c) 3 atm(d) 0.5 atm
›Reveal solutionSolution
Total pressure = 1 atm (air) + rho g h (about 1 atm) = 2 atm. Answer (B).
Absolute pressure at depth h: P = P_atm + rho g h.
…
- CBSE 2025Set ANNUAL1 markMCQQ.Liquid pressure depends upon(a) area of the liquid surface(b) shape of the liquid surface(c) height of the liquid column(d) directions
›Reveal solutionSolution
Hydrostatic pressure is given by P = h * rho * g, which depends only on the depth (height of liquid column) and the liquid's density -- famously independent of the container's shape or cross-sectional area (this is the basis of the hydrostatic paradox).
The pressure at a depth h inside a static liquid of density rho is:
P = h * rho * g
…
- CBSE 2025Set ANN1 markQ.Three vessels A, B and C contain different amounts of liquids are shown. All up to the same height. This phenomenon is an Illustration of ______ .
›Reveal solutionSolution
Liquid stands at the same level in differently shaped connected vessels because static pressure depends only on depth, not on shape or volume - this is the hydrostatic paradox.
For a liquid at rest, the pressure at a depth h below the free surface is P = P0 + rho g h, where P0 is the atmospheric pressure. This shows the pressure depends only on the depth h, the density rho and g - and not on the cross-sectional shape of the container or how much liquid it holds.
…
- CBSE 2024Set ANNUAL1 markMCQQ.Liquid pressure depends upon(a) area of the liquid surface(b) shape of the liquid surface(c) height of the liquid column(d) directions
›Reveal solutionSolution
Pressure inside a liquid at depth h is given by P = hρg — it depends only on the depth (height of liquid column) and the liquid's density, never on the shape or cross-sectional area of the container.
The formula for pressure at a point inside a liquid at rest, at depth h below the free surface, is:
P = h ρ g (+ atmospheric pressure, if considering absolute pressure)
…
- CBSE 2018Set ANNUAL1 markQ.What is pressure head?
›Reveal solutionSolution
Pressure head is pressure expressed as an equivalent height of a fluid column: h = P/(rho g).
In Bernoulli's equation, pressure (P), kinetic energy per unit volume ((1/2) rho v^2), and gravitational potential energy per unit volume (rho g h) are all expressed as energy per unit volume and can each be converted into an equivalent "head" (a height):
Pressure head h = P / (rho g)
…
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