Nano PHYSICS . MECHANICS . WORK AND ENERGY . VARIABLE FORCES AND GRAPHICAL WORK

Variable Forces and Integration

Compute work by integrating force over position in one dimension, and connect that idea to the general force-displacement dot product.

Physics · Mechanics · Work and Energy · Variable Forces and Graphical Work · Variable Forces and Integration
Access for this nano-lesson
Unsigned visitors can show & copy prompts for Steps 1-3. Signed-in free accounts can also Run with AI for Steps 1-2. Paid accounts unlock everything (Steps 1-6 + Help prompts + AI).
Steps 1-3 Free Steps 4-6 Paid
STEP 1
Orient: variable forces and integration
Free
Establish how work is found when force changes with position.
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Goal: define variable-force work using force, displacement, graph area, or integration.
STEP 2
Conceptual grounding: build the variable-force work model
Free
Build a workflow for graph area, integration, signs, and units.
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Goal: identify the correct interval, sign convention, and method for computing work.
STEP 3
Real-world connection: interpret graphs, integrals, and springs
Free
Apply variable-force work reasoning to graphs, springs, friction, gravity, and energy transfer.
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Goal: explain positive, negative, and zero work using signed area and energy transfer.
STEP 4
Check your understanding: diagnostic mini-quiz
Paid
Attempt each question before revealing the variable-force work reasoning.
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Goal: identify and correct area, sign, limit, spring-direction, and unit errors before they become calculation errors.
STEP 5
Practice: solve increasingly rich variable-force work situations
Paid
Apply the model to situations that grow in complexity.
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Goal: write concise, physically justified variable-force work calculations and explanations.