QUICK REVIEW
Quick Review — Static Equilibrium and Stability — Calculus-Based
Refresh the essential ideas and relationships for Static Equilibrium and Stability in just a few minutes.
TIME
5–10 minutes
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A rapid refresh
FINISH WITH
Key ideas refreshed
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Course Alignment
This Physics Sensei Unit Review is an independent learning resource. Use it to reinforce key concepts, prepare for homework, or review before a quiz or exam.
RESOURCE: Physics Sensei Unit Review | UNIT ID: MEC-U23 | TOPIC: Static Equilibrium and Stability | COURSE LEVEL: Calculus-Based
BEST USED ✓ After learning the unit ✓ Before starting homework ✓ Before a quiz or exam
Your Review Plan
Complete these four stages to quickly refresh the essential ideas and confirm you're ready to continue.
4 Stages • Approximately 5–10 minutes.
Quick Recall
Let's quickly refresh what you already know. These short recall activities will help you bring the most important ideas back to mind before reviewing them.
QUICK RECALL
Recall Activity
Complete the three statements from memory before revealing the answer.
State the rigid-body equilibrium conditions in vector form. For a one-coordinate system with potential energy U(θ), what derivative conditions identify an equilibrium and distinguish stable from unstable equilibrium?
Reveal Answers
ΣF = 0 and Στ = 0. Equilibrium: dU/dθ = 0. Stable if d2U/dθ2 > 0; unstable if d2U/dθ2 < 0.
Why it works: The first derivative locates stationary configurations; the second derivative determines local curvature and therefore the response to a small displacement.
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Essential Idea
Take one last look at the most important concept from this unit. If you remember this idea, the rest will come back much more easily.
ESSENTIAL IDEA
Equilibrium and Stability from Forces, Torques, and Energy
Static equilibrium requires ΣF = 0 and Στ = 0. For a generalized coordinate θ, equilibrium also corresponds to dU/dθ = 0. A local minimum of U is stable because a small displacement raises the potential energy and produces a restoring tendency.
ΣF = 0 and Στ = 0. Equilibrium: dU/dθ = 0. Stable if d2U/dθ2 > 0; unstable if d2U/dθ2 < 0.
Example: If U(θ) has a minimum at θ = 0, then small rotations away from zero increase U, so the system tends to return toward θ = 0.
Sensei note: A stationary point of U is not automatically stable; check whether it is a minimum, maximum, or flat higher-order point.
Ready to check your memory?
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Confidence Check
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CONFIDENCE CHECK
Classify the Equilibrium
Answer all three without notes.
At θ = 0, a system has dU/dθ = 0 and d2U/dθ2 > 0. Classify the equilibrium and state why.
Reveal Answers
Stable equilibrium. The positive second derivative means U has a local minimum at θ = 0.
Why it works: Small displacements increase U, producing a restoring force or torque toward the minimum.
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