Calculus I · Unit 3B · lesson
Fluid Pressure and Hydrostatic Force
Learn Fluid Pressure and Hydrostatic Force through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Physics and quantitative applicationsWhat this section is building
Learn Fluid Pressure and Hydrostatic Force through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Work adds force through distance, pumping adds slice weight through lift distance, pressure adds depth-dependent strip force, and marginal or probability models add weighted local contributions.
Draw a coordinate system, define the slice at a general position, express every changing factor in one variable, and state the domain and units.
Confusing mass density with weight density, measuring depth from the wrong reference, or integrating a marginal quantity without an initial value when a total function is requested.
Learning objectives
Integrate pressure over a submerged surface using horizontal strips.
Fluid Pressure and Hydrostatic Force
Pressure depends on depth, force also depends on area
Fluid pressure increases linearly with depth according to , where is weight density. Pressure alone is not force. A thin strip of a submerged plate experiences force approximately equal to pressure at that depth times the strip's area. Integrating these strip forces gives the total hydrostatic force.
The depth variable and strip width depend on the geometry and coordinate choice. A strip near the surface has lower pressure than an equally sized strip deeper down. For vertical plates, horizontal strips usually keep depth constant across each strip and are therefore natural. Check that pressure units multiplied by area units produce force units, and distinguish depth below the surface from height above the bottom.
Fluid pressure at depth is
where is weight density. A horizontal strip of width and thickness has area , so
A rectangular gate
A vertical rectangular gate is 2 m wide and extends from the water surface to depth 3 m. With measured downward,
u3b-fluid-01A 2 m wide rectangular gate extends from the water surface to depth 3 m. Use water weight density 9800 N/m^3. Find the total force.
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Integrate from 0 to 3.
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Read this graph as text
Pressure increasing with depth on a submerged plate. Pressure arrows grow with depth; force on strip equals pressure times strip area. Show horizontal strip, depth, pressure, strip width, and force. Depth is measured below surface, not from tank bottom unless coordinates are converted.
Every relationship in pressure increasing with depth on a submerged plate uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show horizontal strip, depth, pressure, strip width, and force.
Pressure increasing with depth on a submerged plate. Show horizontal strip, depth, pressure, strip width, and force.
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