Potential Energy and Energy Diagrams

Energy Diagrams: Ball in Free Fall

Energy diagrams are graphical representations that help visualize how energy is distributed and transformed in a physical system. For a ball in free fall, the total mechanical energy remains constant if only conservative forces (like gravity) act on the system.

• Mechanical Energy (Emech): The sum of kinetic energy (K) and potential energy (U).

• Conservation of Mechanical Energy:

• Gravitational Potential Energy: (where y is the height above a reference point)

• Kinetic Energy:

Example: As a ball falls from height to , decreases while increases, but remains constant.

Energy Diagram for a Mass-Spring System

For a mass attached to a spring, the energy diagram shows how energy oscillates between kinetic and potential forms as the mass moves.

• Spring Potential Energy:

• Kinetic Energy:

• Equilibrium Position: The point where the spring is neither stretched nor compressed ().

• At maximum displacement: All energy is potential ().

• At equilibrium: All energy is kinetic ().

Example: As the mass oscillates, energy is exchanged between and , but remains constant.

Equilibrium and Stability

Interpreting Energy Diagrams

Energy diagrams can be used to identify equilibrium positions and their stability.

• Total Energy:

• Kinetic Energy:

• Equilibrium Positions: Points where the potential energy has a local minimum or maximum.

• At equilibrium: The force is zero ().

Example: At positions , , (local minima or maxima of ), the particle is at equilibrium. The nature of equilibrium (stable or unstable) depends on the curvature of at these points.

Types of Equilibrium

• Stable Equilibrium: Occurs at a local minimum of . A small displacement results in a restoring force back toward equilibrium.

• Unstable Equilibrium: Occurs at a local maximum of . A small displacement results in a force away from equilibrium.

Example: If a particle at (local minimum) is given a small kick, it oscillates about (stable). If at (local maximum), a small kick moves it away (unstable).

Force and Potential Energy

Relationship Between Force and Potential Energy

The force associated with a potential energy function is given by the negative gradient (slope) of the potential energy with respect to position.

• Work-Energy Relation:

• Force from Potential Energy:

Example: For a spring, , so (Hooke's Law).

Equilibrium Conditions

• Equilibrium: Occurs when (force is zero).

• Stability: If , equilibrium is stable; if , equilibrium is unstable.

Conservative and Nonconservative Forces

Definitions and Properties

• Conservative Force: A force for which the work done moving a particle between two points is independent of the path taken.

• Nonconservative Force: A force for which the work done depends on the path (e.g., friction).

• Potential Energy: Can only be defined for conservative forces.

Mathematical Condition:

• For to be well-defined, must be path-independent.

The Energy Principle Revisited

Generalized Energy Conservation

When both conservative and nonconservative forces are present, the total energy of a system is accounted for by including work done by external and nonconservative forces.

• System: Particle + environment (with possible dissipation)

• Energy Conservation Equation:

• : Change in thermal energy (due to dissipation)

• : Change in kinetic energy

• : Change in potential energy

• : Work done by external forces

• : Work done by conservative forces

• : Work done by nonconservative forces

Final Form of Energy Conservation:

Example: In a system with friction, accounts for energy lost as heat.

Summary Table: Types of Equilibrium

Additional info: These notes cover material from Chapter 10: Interactions and Potential Energy, including the relationship between force and potential energy, energy diagrams, equilibrium, and the distinction between conservative and nonconservative forces.