Potential Energy and Conservation of Energy

Introduction

This chapter explores the concepts of potential energy and the conservation of energy in physical systems. It distinguishes between conservative and nonconservative forces, explains how work and energy are related, and introduces the mathematical framework for analyzing energy transformations.

Conservative and Nonconservative Forces

Types of Forces

Forces in physics are classified based on whether the work they do can be fully recovered as mechanical energy.

• Conservative Forces:

  • Work and energy associated with these forces can be recovered.

  • The work done is stored as energy that can be released as kinetic energy at a later time.

  • Examples: Gravity, spring force.

  • Work done by a conservative force is path independent; it depends only on the initial and final positions.

  • The work done by a conservative force over a closed path is zero.

• Nonconservative Forces:

  • These forces are generally dissipative; work done against them cannot easily be recovered.

  • Examples: Friction force, drag.

  • Work done by a nonconservative force is path dependent.

  • Energy is often transformed into other forms, such as heat.

Work Done by Gravity

Gravitational Work Near Earth's Surface

Gravity is a conservative force. The work it does on an object depends only on the change in vertical position.

• Net force on a falling object:

• Work done by gravity: Where is the change in vertical position.

• If , then , making positive (force and displacement are in the same direction).

Potential Energy

Definition and Gravitational Potential Energy

Potential energy is energy associated with the position of an object within a system. For every conservative force, a potential energy function can be defined.

• General definition:

• Gravitational Potential Energy:

• Choosing as the reference point,

• Interpretation: Gravitational potential energy is the energy due to an object's position in Earth's gravitational field.

Conservation of Mechanical Energy

Principle of Conservation

In the absence of nonconservative forces, the total mechanical energy of a system remains constant.

• Mechanical Energy: Where is kinetic energy and is potential energy.

• Conservation equation:

• Energy can transform between kinetic and potential forms, but the total remains unchanged.

Elastic Potential Energy

Spring Systems

Elastic potential energy is stored in a stretched or compressed spring.

• Reference position: (equilibrium position).

• Spring potential energy: Where is the spring constant and is the displacement from equilibrium.

• This energy can be converted to kinetic energy when the spring returns to equilibrium.

Conservation of Energy Including a Spring

Energy in Systems with Springs

When a spring is involved, its potential energy is included in the total mechanical energy.

• Total mechanical energy:

• The same conservation principles apply: energy can shift between kinetic, gravitational, and elastic forms.

Work Done by Nonconservative Forces

Effects on Mechanical Energy

Nonconservative forces, such as friction, cause mechanical energy to change, often transforming it into other forms like heat.

• General energy equation:

• Or,

• Positive nonconservative work increases total mechanical energy; negative work decreases it.

Potential Energy Curves

Graphical Representation

Potential energy curves plot as a function of position, helping visualize how energy changes in a system.

• Example: The shape of a hill or roller coaster can be represented by a gravitational potential energy curve.

• These curves are useful for analyzing stability and equilibrium points in physical systems.

Summary Table: Conservative vs. Nonconservative Forces

Key Equations

• Work by gravity:

• Gravitational potential energy:

• Spring potential energy:

• Conservation of mechanical energy:

• General energy change with nonconservative work:

Applications and Examples

• Trampoline jump: Elastic potential energy in the spring converts to kinetic energy as the jumper rises.

• Bungee jump: Gravitational potential energy converts to elastic potential energy in the cord, then back to kinetic energy.

• Roller coaster: Mechanical energy shifts between gravitational potential and kinetic energy as the coaster moves along the track.

Additional info: The images in the slides (trampoline, bungee jump) illustrate real-world applications of energy conservation and transformation between kinetic, gravitational, and elastic potential energy.