Calculus I · Unit 3B · lesson

Work Done on Springs

Concept

Learning objectives

Use Hooke’s law and integrate variable spring force.

Work Done on Springs

Explanation

A spring's force changes with extension

Hooke's Law models an ideal spring by $F(x)=kx$ , where $x$ is displacement from the natural length and $k$ 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 $x$ . 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 $k$ , displacement, and work.

Hooke's law gives $F(x)=kx$ , where $x$ is displacement from equilibrium. Work to stretch from $a$ to $b$ is

W=\int_a^bkx\,dx=\frac{k}{2}(b^2-a^2).

Worked example

Calibrate the spring first

A 20-newton force stretches a spring $0.10$ meter. Then $k=20/0.10=200$ N/m. Work to stretch from equilibrium to $0.25$ meter is

\int_0^{0.25}200x\,dx=6.25\text{ J}.

Interactive checku3b-spring-01

A spring has $k=200$ N/m. Find the work to stretch it from 0 to 0.25 m.

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Show hint

Integrate $200x$ from 0 to 0.25.

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After the explanation

Use the section idea

Reading lens

Translate the situation into rate, density, force, pressure, or probability before calculating; the integral is the final accumulation step, not the first modeling decision.

Mental model

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.

Decision

Draw a coordinate system, define the slice at a general position, express every changing factor in one variable, and state the domain and units.

Common trap

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.

Check yourself

Does the setup respond correctly when the slice moves, and can a units or scale check expose a missing factor?

Source & rights

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