Chapter 10: Interactions and Potential Energy - Key Concepts

Potential Energy

• Potential energy, $U$, is the energy stored in an interaction inside a system.
• Change in potential energy: $\Delta U = -W_{int}$
• Forces can act across system boundaries (work) or within the system (potential energy).

Gravitational Potential Energy

• Gravitational potential energy $U_G$ is the interaction energy between two masses.
• Near Earth's surface: $U_G = mgy + C$ (C is any constant).
• Work with it by choosing a reference point where $U_G = 0$.
• What matters is $\Delta U<em>G = U</em>{G,final} - U_{G,initial}$.
• $\Delta U_G$ is independent of the reference point.

Mechanical Energy

• $E<em>{mech} = K</em>{tot} + U_{tot}$
• For an object-Earth system, if $W<em>{ext} = 0$ and no dissipative forces: $K</em>i + U<em>{Gi} = K</em>f + U_{Gf}$
• $\Delta E<em>{sys} = \Delta K + \Delta U</em>G$
• Thermal and chemical energy not included in "mechanical energy."
• When mechanical energy is not conserved we have: $\Delta E<em>{sys} = \Delta K + \Delta U</em>G + \Delta E_{th} = 0$
• Or the Energy Conservation statement is: $K<em>i + U</em>{Gi} = K<em>f + U</em>{Gf} + \Delta E_{th}$

Elastic Potential Energy

• Elastic potential energy: $U_{Sp} = \frac{1}{2} k (\Delta s)^2$
• For an isolated system with gravitational and elastic potential energy we have: $\Delta E<em>{sys} = \Delta K + \Delta U</em>G + \Delta U<em>{Sp} + \Delta E</em>{th} = 0$
• Or: $K<em>i + U</em>{Gi} + U<em>{Sp i} = K</em>f + U<em>{Gf} + U</em>{Sp f} + \Delta E_{th}$

Conservation of Energy

• Total energy of an isolated system is constant: $E<em>{sys} = K + U + E</em>{th}$
• Mechanical energy is conserved if the system is isolated and non-dissipative.

Energy Diagrams

• Potential energy curve (PE) is determined by system properties.
• Total energy curve (TE) can be changed by initial conditions.
• A turning point is where $K = 0$.
• A particle cannot be at positions with U > E.
• Stable equilibrium: local minimum.
• Unstable equilibrium: local maximum.

Force and Potential Energy

• $F_s = -\frac{dU}{dx}$
• The interaction force is the negative slope of the potential energy curve.

Conservative and Nonconservative Forces

• Conservative force: work done is independent of the path.
• Potential energy can be associated with conservative forces.
• Examples: Gravity, Springs, Electrostatic.
• Nonconservative forces: Friction, Air Resistance.

The Energy Principle Revisited

• $\Delta K + \Delta U + \Delta E<em>{th} = \Delta E</em>{mech} + \Delta E<em>{th} = \Delta E</em>{sys} = W_{ext}$
• $K<em>i + U</em>i + W<em>{ext} = K</em>f + U<em>f + \Delta E</em>{th}$