Potential Energy

Potential energy \(U\) is defined relative to a chosen zero; only changes \(\Delta U\) matter physically.

• Why “zero” is arbitrary
• What measurements depend on
• Energy bookkeeping

A force can be described by a function \(U(x)\) so energy changes replace force details.

• \(U(x)\) as an “energy landscape”
• Reading changes from graphs
• Turning points preview

In 1D: \(F_x=-\frac{dU}{dx}\). In 3D: \(\vec F=-\nabla U\).

• Why the minus sign appears
• Slope gives force direction
• Equilibrium points preview

Use \(U(x)\) to see where motion is allowed and where it turns around.

• Allowed regions: \(K \ge 0\)
• Turning points: \(K=0\)
• Qualitative motion prediction

Convert between force and \(U(x)\), read energy graphs, and predict motion.

• Choose reference level drills
• From \(U(x)\) to \(F(x)\) (slopes)
• Turning-point identification sets
• Allowed region questions
• Concept checks (signs & meaning)

When a force is conservative, its work can be written as \(W=-\Delta U\), enabling conservation of energy.

• Path independence idea
• Energy conservation setup
• Next topic connection