Calculus I · Unit 3B · lesson
Work and Variable Force
Learn Work and Variable 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 Work and Variable 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
Model work as the integral of force over displacement and interpret units.
Work and Variable Force
Work accumulates force through distance
For a constant force acting in the direction of motion, work is force times distance. If the force varies with position, divide the displacement into short intervals. On each interval, the force is approximately constant, so the local work is . The integral is the limiting total.
Signs and units carry physical meaning. Force in the direction of displacement contributes positive work, while an opposing force contributes negative work. In SI units, newtons times meters give joules. A graph of force versus position makes the interpretation visible: work is signed area under the force curve, not area under a position or time graph unless the variables have been transformed appropriately.
For constant force parallel to motion, . For varying force,
In SI units, newtons times meters gives joules.
A linearly increasing force
A force newtons moves an object from to meters:
u3b-work-01A force N moves an object from 0 to 4 m. Find the work.
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Integrate force with respect to displacement.
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Read this graph as text
Work as force-distance accumulation. Area under a force-position graph has units newton-meters and equals work. Show force curve and thin work contribution F(x)dx. Do not graph force versus time for a force-distance work integral without a change of variables.
Every relationship in work as force-distance accumulation uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show force curve and thin work contribution F(x)dx.
Work as force-distance accumulation. Show force curve and thin work contribution F(x)dx.
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