M091 · Fluid mechanics

Hydrostatics & Hydraulic Press

Move a pressure probe through cylindrical or flared reservoirs. Operate connected pistons and audit the force, stroke and gravitational energy trade-off; infer density from acquired pressures.

Probe gauge pressure $\text{Ready}$

Bottom gauge pressure $\text{Ready}$

Probe height above bottom $\text{Ready}$

Liquid volume $\text{Ready}$

Liquid weight $\text{Ready}$

Floor force on liquid $\text{Ready}$

Wall force on liquid $\text{Ready}$

Piston area ratio $\text{Ready}$

Upward output force $\text{Ready}$

Output stroke $\text{Ready}$

Input work $\text{Ready}$

Output work $\text{Ready}$

Fluid gravity energy gain $\text{Ready}$

Work budget residual $\text{Ready}$

Acquired stations $\text{Ready}$

Density from raw data $\text{Ready}$

Fit coordinate residual $\text{Ready}$

Approximation status $\text{Ready}$

GEOMETRY / PROBE / ACQUIRE

Does a larger force mean more work?

Apparatus: reservoir / press

b

0.00\,

Tank bottom radius

R_0

0.30\,m

Wall flare

f

0.00\,

Liquid height

z_0

1.20\,m

Applied surface gauge pressure

p_s

0.00\,kPa

Input piston radius

r_1

4.00\,cm

Output piston radius

r_2

12.00\,cm

Input force

F_1

150.00\,N

Prescribed input stroke

s_1

0.10\,m

Probe height fraction

z_p/z_{\mathrm{available}}

0.50\,

Reference fluid density

\rho

1000.00\,kg/m^3

Gravity

g

9.81\,m/s^2

Pressure noise bound

\epsilon_p

0.00\,kPa

Uniform density, constant gravity and gauge pressure relative to ambient. Cylinder or linearly flared radius; a nonzero surface pressure uses a massless fitted cap. Side-wall force completes the vertical fluid balance. No cap mass, capillarity, sloshing, seal friction or fluid compressibility.

The press has equal initial piston heights, massless pistons and an ideal liquid-filled connection. Input stroke is downward, output stroke upward. Pressure varies with elevation. Input work equals output work plus the liquid gravity energy gain. A negative output force needs a downward balancing force and is not acquired as a lifting experiment. This is prescribed quasi-static motion, not acceleration.

Acquire gauge pressure at known depths; least squares with a free intercept infers density using known gravity. The reference density is excluded from the fit and CSV. Moving the probe preserves observations; apparatus, geometry, fluid, force, stroke or noise edits clear them. Repeated stations replace old readings. Noise is reproducible and bounded, not independent repeats.

Uniform incompressible liquid, quasi-static balance. Actual metre geometry; fixtures are schematic. Sensor observations are simulated. Prescribed piston strokes are balanced states, not a dynamic lift.

Start with the essentials

Focus question: Does a larger force mean more work?
One-sentence intuition: Depth governs pressure; area governs force and reciprocal travel.

Core mathematical model

Hydrostatic profile

p(z)=p_s+\rho g(z_s-z)

Depth sets pressure; vessel shape does not.

Conserved piston volume

A_1s_1=A_2s_2

Downward input stroke raises the output face.

Unequal piston heights

F_2=A_2\!\left[\frac{F_1}{A_1}+\rho g(z_1-z_2)\right]

Force multiplication has a height correction.

The missing work goes into gravity

W_1-W_2=\frac{\rho g A_1s_1^2}{2}\!\left(1+\frac{A_1}{A_2}\right)

The two columns start at the same height; their gravity energy increases as the faces separate.

Density from acquired pressure

p_i=\alpha d_i+\beta,\qquad\widehat\rho=\widehat\alpha/g

Known depths and gravity turn a fitted pressure slope into density. The intercept absorbs surface pressure.

Common difficulties

Acquire before inferring

Typical misconceptionThe estimate repeats a reference slider.

Better mental modelClear the observations: the estimate disappears. Add noise or change the reference parameter, then acquire again.

Know the approximation

Typical misconceptionA smooth apparatus proves every displayed state is physically reliable.

Better mental modelRead the domain status; extrapolated states remain visible but cannot be acquired.

Run the experiment

01
Change the vessel
Compare cylindrical and flared presets at equal height.
What to observe: Bottom pressure stays fixed; wall forces account for the extra weight.
02
Move and acquire
Drag the gold probe track, then acquire a depth profile.
What to observe: The fitted pressure gradient yields density.
03
Operate the press
Select the press preset and increase input stroke.
What to observe: Output travel shrinks while gravity consumes part of the work.
04
Test the head correction
Select elevation matters and inspect the work residual.
What to observe: The equal-height force reference diverges as piston heights separate.