Potential Energy and Energy Conservation

Learning Outcomes

This chapter explores the concepts of potential energy and energy conservation in mechanical systems. Students will learn to:

• Apply gravitational potential energy to vertical motion problems.

• Use elastic potential energy in systems involving springs or elastic objects.

• Distinguish between conservative and nonconservative forces.

• Interpret energy diagrams to analyze motion under conservative forces.

Introduction to Energy Concepts

Energy can be stored and transformed between different forms. For example, as a sandhill crane descends, its potential energy is converted to kinetic energy. Understanding these transformations is key to analyzing physical systems.

Gravitational Potential Energy

Definition and Formula

When a particle is in Earth's gravitational field, it possesses gravitational potential energy:

• Formula:

• m: mass of the particle

• g: acceleration due to gravity

• y: vertical position

As an object descends, its gravitational potential energy decreases and is converted to kinetic energy, increasing its speed.

Work and Gravitational Potential Energy

• The change in gravitational potential energy is equal to the work done by gravity.

• When an object moves downward ( decreases), gravity does positive work and decreases.

• When an object moves upward ( increases), gravity does negative work and increases.

The Conservation of Mechanical Energy

Principle of Conservation

The total mechanical energy of a system is the sum of its kinetic and potential energies:

• Formula:

• A conserved quantity remains constant if only conservative forces act.

• If only gravity does work, mechanical energy is conserved:

Example: A high jumper's mechanical energy remains constant during flight if air resistance is neglected.

Forces Other Than Gravity

• When non-gravitational forces (e.g., air resistance) do work, mechanical energy is not conserved.

• Energy is transformed into other forms, such as heat or internal energy.

Work and Energy Along a Curved Path

The expression for gravitational potential energy applies regardless of the path's shape:

• Formula:

• Only the vertical displacement matters for gravitational work.

Elastic Potential Energy

Definition and Formula

An object is elastic if it returns to its original shape after deformation. Elastic potential energy is stored in objects like springs:

• Formula:

• k: spring constant

• x: displacement from equilibrium (positive if stretched, negative if compressed)

Example: The Achilles tendon acts like a spring, storing and releasing energy during running.

Work Done by a Spring

• When a spring is stretched or compressed, it does work on attached objects.

• Work is positive when the spring returns to equilibrium, negative when stretched or compressed further.

Graphical Representation

• The graph of for an ideal spring is a parabola.

• Elastic potential energy is always non-negative.

Situations with Both Gravitational and Elastic Forces

When both forces are present, total potential energy is the sum:

• Formula:

Example: A person jumping on a trampoline experiences both gravitational and elastic potential energies.

Conservative and Nonconservative Forces

Conservative Forces

• Allow conversion between kinetic and potential energy.

• Examples: gravity, spring force.

• Work done depends only on endpoints, not the path.

• Properties:

  • Can be expressed as a potential energy function.

  • Reversible.

  • Path-independent.

  • Zero work for closed paths.

Nonconservative Forces

• Do not store potential energy; convert mechanical energy to other forms (e.g., heat).

• Examples: friction, air resistance.

• Work done depends on the path taken.

• Dissipative forces are a type of nonconservative force.

Example: Internal friction in a rolling tire increases its temperature and pressure.

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed:

• Formula:

• Nonconservative forces change the internal energy of a system.

Force and Potential Energy

One Dimension

• A conservative force can be derived from its potential energy function:

• Large changes in correspond to large force magnitudes.

• Force acts to decrease potential energy.

Elastic and Gravitational Forces as Functions

• Elastic: ,

• Gravitational: ,

Three Dimensions

• Components of a conservative force are given by partial derivatives:

• The vector form (gradient):

Physical Interpretation

• Regions with steep gradients in correspond to strong forces.

• Force pushes systems toward lower potential energy.

• Example: A hiker is pushed downhill by gravity toward lower .

Energy Diagrams

An energy diagram plots the potential energy function and the total mechanical energy .

• Shows limits of motion and equilibrium points.

• Helps visualize how energy is distributed and transformed.

• Example: A glider attached to a spring oscillates between points where equals .

Equilibrium and Stability

Force and Potential Energy Graphs

• At points where , force is zero (equilibrium).

• If is at a minimum, equilibrium is stable (restoring force).

• If is at a maximum, equilibrium is unstable (force pushes away).

Unstable Equilibrium

• Objects in unstable equilibrium will move away from their position if slightly disturbed.

• Example: Acrobats balancing on unicycles are in unstable equilibrium; any tip causes them to fall further.

Additional info: These notes expand on the textbook slides by providing definitions, formulas, and examples for each concept, ensuring a self-contained study guide for exam preparation.