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 UG = U{G,final} - U_{G,initial}.
  • \Delta U_G is independent of the reference point.

Mechanical Energy

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

Conservation of Energy

  • Total energy of an isolated system is constant: E{sys} = K + U + E{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{th} = \Delta E{mech} + \Delta E{th} = \Delta E{sys} = W_{ext}
  • Ki + Ui + W{ext} = Kf + Uf + \Delta E{th}