Calculus I · Unit 3B · lesson
Work Done on Springs
Learn Work Done on Springs 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 Done on Springs 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
Use Hooke’s law and integrate variable spring force.
Work Done on Springs
A spring's force changes with extension
Hooke's Law models an ideal spring by , where is displacement from the natural length and is the spring constant. Because the required force grows as the spring is stretched or compressed farther, work cannot be found by multiplying one endpoint force by the entire distance. The force must be integrated over the displacement interval.
Be careful about the coordinate used for . If lengths are measured from a wall or from the spring's total length, first convert them to extension from equilibrium. The work done by an external agent and the work done by the spring have opposite signs, depending on the convention. State which quantity is being computed and include consistent units for , displacement, and work.
Hooke's law gives , where is displacement from equilibrium. Work to stretch from to is
Calibrate the spring first
A 20-newton force stretches a spring meter. Then N/m. Work to stretch from equilibrium to meter is
u3b-spring-01A spring has N/m. Find the work to stretch it from 0 to 0.25 m.
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Integrate from 0 to 0.25.
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