BackCH7-EXAM-2-Potential Energy and Energy Conservation: Gravitational Potential Energy and the Work-Energy Theorem
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Chapter 7: Potential Energy and Energy Conservation
Introduction to Conservative Forces and Potential Energy
In this chapter, we explore the concept of conservative forces and their relationship to potential energy. We also introduce the work-energy theorem and discuss how mechanical energy is conserved in systems influenced only by conservative forces.
Conservative Force: A force for which the work done is independent of the path taken and depends only on the initial and final positions.
Potential Energy: The energy associated with the position or configuration of an object in a force field (e.g., gravitational, elastic/spring).
Work-Energy Theorem: The total work done on an object by all forces equals the change in its kinetic energy.
Total Mechanical Energy: The sum of an object's kinetic and potential energies.
For this course, the main conservative forces considered are gravity and the elastic/spring force.
Gravitational Potential Energy
Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. For an object of mass m at a vertical height y above a reference point in a uniform gravitational field g:
Formula:
Gravitational potential energy is a specific form of potential energy associated with the force of gravity.
The work done by gravity and the change in gravitational potential energy are related by:
Calculating Work Done by Gravity
To calculate the work done on an object by gravity, consider the direction of displacement relative to the force:
When displacement is downward (), gravity does positive work and potential energy decreases.
When displacement is upward (), gravity does negative work and potential energy increases.
General formula for work done by gravity:
This result is path independent: only the vertical change matters, not the actual path taken.
Example: If a ball falls from to , the work done by gravity is .
Potential Energy and Path Independence
The concept of potential energy simplifies calculations for conservative forces. For gravity, the work done depends only on the vertical displacement between two points, not the path taken.
Potential energy allows us to avoid complex path integrals.
For gravity: for any path.
The Work-Energy Theorem and Mechanical Energy Conservation
The work-energy theorem states that the total work done on an object equals the change in its kinetic energy:
Total work can be split into work by gravity and work by other forces:
When only conservative forces do work, mechanical energy is conserved:
Mechanical Energy Conservation:
If other forces (non-conservative) do work, then:
Example: A climber pulls a supply pack up a slope at constant speed. The work done by gravity, normal force, and the climber can be analyzed using the work-energy theorem and mechanical energy conservation.
Summary Table: Key Relationships
Quantity | Formula | Description |
|---|---|---|
Gravitational Potential Energy | Energy due to position in a gravitational field | |
Work by Gravity | Work done by gravity over vertical displacement | |
Mechanical Energy | Sum of kinetic and potential energies | |
Work-Energy Theorem | Total work equals change in kinetic energy | |
Energy Conservation (Conservative Forces Only) | Mechanical energy remains constant |
Key Points
Conservative forces allow the definition of potential energy and ensure mechanical energy conservation.
Gravitational potential energy depends only on vertical position.
Work done by gravity is path independent and relates directly to changes in potential energy.
Mechanical energy is conserved when only conservative forces act; otherwise, non-conservative work must be included.
