Potential Energy and Energy Conservation

Introduction to Energy in Mechanics

In mechanics, energy is a fundamental concept that comes in two primary forms: kinetic energy and potential energy. Kinetic energy, denoted as K, is the energy of motion, while potential energy, denoted as U, is the energy of position or configuration.

• Kinetic Energy (K): Energy due to an object's motion.

• Potential Energy (U): Energy stored due to an object's position or arrangement.

The General Definition of Work

Work in Physics

Work is a measure of energy transfer that occurs when a force acts upon an object to cause displacement. The general definition of work depends on whether the force is constant or variable.

• For constant forces in 3 dimensions:

• For a non-constant force in 1 dimension (along the x-axis):

• For a variable force in 3 dimensions (line integral):

This means you must integrate the force along the path taken from point A to point B. The result may depend on the path unless the force is conservative.

Conservative and Non-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. Examples: gravity, spring force.

• Non-Conservative Force: A force for which the work done depends on the path. Example: friction.

Potential Energy and Conservative Forces

Definition of Potential Energy

For every conservative force, there exists a potential energy function, U, such that:

• The negative sign indicates that when a conservative force does positive work, the system's potential energy decreases.

• Types of potential energy include gravitational, spring (elastic), electric, magnetic, chemical, and nuclear.

• There is no potential energy function for friction, as it is non-conservative.

Gravitational Potential Energy

Definition and Formula

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically near the Earth's surface.

Defining gravitational potential energy:

• m: mass of the object (kg)

• g: acceleration due to gravity ( near Earth's surface)

• y: vertical position relative to a chosen reference point

The value of potential energy depends on the choice of the reference point (where ). Usually, the lowest point in the system is chosen so that is non-negative.

Example

• Problem: A 0.75-kg flower pot is on a balcony 50 m above the street. What is its potential energy (a) relative to the street, (b) relative to the balcony?

• Solution: Use with measured from the chosen reference point.

Key Point

• Only changes in potential energy () are physically meaningful, and these are independent of the coordinate system.

Example

• Problem: A 0.75-kg flower pot falls from a balcony 50 m above the street. Find the change in gravitational potential energy from the balcony to 10 m above the ground.

• Solution:

Path Independence of Potential Energy

Concept and Example

The change in gravitational potential energy is independent of the path taken between two points; it depends only on the initial and final positions.

• Example: Two paths lead to the top of a hill, one steep and direct, the other long and winding. The change in potential energy is the same for both paths.

This property is true for all types of potential energy associated with conservative forces.

Conservation of Mechanical Energy

Conservative Systems

A conservative system is one in which only conservative forces are acting. In such systems, the total mechanical energy (sum of kinetic and potential energy) is conserved.

Thus, the total mechanical energy remains constant:

• Energy can transform between kinetic and potential forms, but the total remains unchanged in the absence of non-conservative forces.

Summary Table: Conservative vs. Non-Conservative Forces

Key Takeaways

• Work is the energy transferred by a force acting over a distance.

• Conservative forces allow the definition of potential energy and lead to energy conservation in isolated systems.

• Gravitational potential energy is given by near Earth's surface.

• Only changes in potential energy are physically meaningful and are path-independent for conservative forces.

• Mechanical energy (kinetic + potential) is conserved in systems with only conservative forces.