Nano PHYSICS . MECHANICS . WORK AND ENERGY . POTENTIAL ENERGY FUNCTIONS

Force as the Gradient of U

In one dimension, force is the negative derivative of potential energy. In three dimensions, force is the negative gradient of U.

Physics · Mechanics · Work and Energy · Potential Energy Functions · Force as the Gradient of U
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: force as the gradient of u
Free
Establish potential energy as a position-dependent energy-bookkeeping tool.
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Goal: define potential energy using reference choice, changes, graph meaning, and units.
STEP 2
Conceptual grounding: build the potential-energy model
Free
Build a workflow for reference choices, graphs, slopes, allowed regions, and turning points.
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Goal: connect U(x) graphs to energy changes, force direction, allowed regions, and turning points.
STEP 3
Real-world connection: interpret energy graphs and forces
Free
Apply potential-energy reasoning to gravity, springs, energy landscapes, and conservative-force previews.
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Goal: explain reference choice, slope-force relations, potential wells, and conservative-force previews in physical contexts.
STEP 4
Check your understanding: diagnostic mini-quiz
Paid
Attempt each question before revealing the potential-energy reasoning.
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Goal: identify and correct reference, slope-sign, graph-reading, and allowed-region errors before they become calculation errors.
STEP 5
Practice: solve increasingly rich potential-energy situations
Paid
Apply the model to situations that grow in complexity.
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Goal: write concise, physically justified potential-energy-function explanations and solutions.
STEP 6
Summary & reflection + explorations
Paid
Consolidate the model and explore changing references, graphs, slopes, and conservation statements.
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🔒 ▶ Explore interactive simulation
Goal: carry the model forward into conservative forces, mechanical-energy conservation, and qualitative motion prediction.