SCIENCE · INVESTIGATE AND EXPLAIN

Account for hydraulic force gain without inventing energy

An increase in pressure applied to a confined ideal fluid is transmitted through it. At equal piston elevations the same gauge-pressure increase acts on both piston faces. Force equals pressure times area, so a larger output piston can balance a larger force. The transmitted quantity is pressure, not identical total force.

Learning outcome: Hold force fixed while doubling output area, then hold area fixed while changing input force. Retain pressure, displaced volume, both forces, strokes and works; explain why neither comparison creates energy.

Starting recap

Declare density, pressure reference, flow regime and admissible force constraints. Convert area and volume units before balancing forces, mass or energy.

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Try

At output area 40 cm² and input force 10 N, force gain is 20, output force 200 N and output stroke 0.005 m. Both works are 1 J.

Open the experiment

Predict

An ideal hydraulic device doubles output piston area at fixed input area and stroke. The output stroke:

Your prediction and reason

Experiment

Hold force fixed while doubling output area, then hold area fixed while changing input force. Retain pressure, displaced volume, both forces, strokes and works; explain why neither comparison creates energy.

Trade force for displacement without creating work

Worked teaching model · synthetic values · Ideal incompressible fluid, equal piston elevations, massless frictionless pistons. Input area 2 cm² and input stroke .1 m. Quasistatic balanced load; atmospheric pressure cancels on exposed faces.

Output piston area: 20 cm² Input force: 20 N

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Output displacement for an input stroke

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: volume continuity (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Output displacement for an input stroke

volume continuity

input stroke (m)	output stroke (m)
0	0
0.0025	0.00025
0.005	0.0005
0.0075	0.00075
0.01	0.001
0.0125	0.00125
0.015	0.0015
0.0175	0.00175
0.02	0.002
0.0225	0.00225
0.025	0.0025
0.0275	0.00275
0.03	0.003
0.0325	0.00325
0.035	0.0035
0.0375	0.00375
0.04	0.004
0.0425	0.00425
0.045	0.0045
0.0475	0.00475
0.05	0.005
0.0525	0.00525
0.055	0.0055
0.0575	0.00575
0.06	0.006
0.0625	0.00625
0.065	0.0065
0.0675	0.00675
0.07	0.007
0.0725	0.00725
0.075	0.0075
0.0775	0.00775
0.08	0.008
0.0825	0.00825
0.085	0.0085
0.0875	0.00875
0.09	0.009
0.0925	0.00925
0.095	0.0095
0.0975	0.00975
0.1	0.01

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Gauge pressure increase	100000	Pa
Force multiplication ratio	10	ratio
Balanced output force	200	N
Output stroke	0.01	m
Input work	2	J
Output work	2	J
Displaced fluid volume	0.00002	m³

Equal pressure increase can multiply force, but the larger piston travels less.

Pascal transmission gives Fout/Aout=Fin/Ain. Incompressible volume conservation gives Ain xin=Aout xout, so ideal work is equal. This quasistatic packet does not predict acceleration, friction, leakage or structural safety. Different elevations would add hydrostatic terms.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Equal pressure increase can multiply force, but the larger piston travels less.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Declare the fluid, reference and evidence

Build the algebraic model

The input area is 2 cm² and its prescribed stroke is 0.1 m. If output area is 20 cm², the force multiplication is ten. With 20 N input force, the pressure increase is 100000 Pa and balanced output force is 200 N. These are quasistatic force balances for massless frictionless pistons, not acceleration predictions.

Check the original numerical packet

Incompressibility supplies the missing displacement relation: input area times input stroke equals output area times output stroke. Here 2×10⁻⁴×0.1=2×10⁻⁵ m³ is displaced, so the output moves only 0.01 m. Input work 20×0.1=2 J equals output work 200×0.01=2 J. A gain in force is paid for by a reduction in distance.

Challenge the assumptions and transfer

The apparatus is a virtual idealization, not a construction or safe-load recipe. Leakage, friction, piston mass, compressibility and different elevations would change the account. Atmospheric forces cancel only because exposed faces share the same atmospheric reference. For unequal elevations, the fluid pressure difference must include a hydrostatic term rather than simply copying the equal-elevation result.

Worked original case

At output area 40 cm² and input force 10 N, force gain is 20, output force 200 N and output stroke 0.005 m. Both works are 1 J.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

Input area is 3 cm² and output area 15 cm². An ideal input stroke is 0.2 m. Find output stroke in metres.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All records are supplied or generated educational evidence. No pressure rigs, liquid handling, bodily measurements or physical exposure protocols are requested. Virtual admissibility does not certify real equipment safety, practical skill or independent mastery.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which assumptions permit pressure equality and which permit volume conservation?

Your explanation and limitations

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Previous: Track gauge pressure across a density interface Next: Reject an impossible cord force before reading apparent weight

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