Potential Energy and Energy Conservation

Introduction to Energy in Mechanics

In classical mechanics, energy is a fundamental concept that describes the ability of a system to perform work. There are two primary types of energy: kinetic energy and potential energy. Kinetic energy, denoted as , is the energy of motion, while potential energy, denoted as , is the energy of position or configuration.

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

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

The General Definition of Work

Work is a measure of energy transfer that occurs when a force acts upon an object to cause displacement. The definition of work varies depending on whether the force is constant or variable, and whether the motion occurs in one or multiple dimensions.

• Work for Constant Force in 3 Dimensions:

  • Equation:

  • Where is the force vector and is the displacement vector.

• Work for Non-Constant Force in 1 Dimension:

  • Equation:

  • Where is a function of position .

• Work for Variable Force in 3 Dimensions (Line Integral):

  • Equation:

  • Where the path from to may be curved, and is evaluated along the path.

  • Expanded:

Conservative Force: A force for which the work done is independent of the path taken, depending only on the initial and final positions. Examples include gravity and spring force.

Potential Energy and Conservative Forces

For every conservative force, there exists a corresponding potential energy function. The work done by a conservative force is related to the change in potential energy:

• Equation:

• 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, electric, magnetic, chemical, and nuclear.

• Non-conservative forces (e.g., friction) do not have associated potential energies, as their work depends on the path taken.

Gravitational Potential Energy

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

• Change in gravitational potential energy:

• Definition near Earth's surface:

  • Where is mass, is acceleration due to gravity, and is height above a chosen reference point.

• The reference point () is arbitrary; usually, the lowest point in the system is chosen so is non-negative.

Example: Rank the gravitational potential energies of balls at different heights. The higher the ball, the greater its .

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

Key Point: Only changes in potential energy () are physically relevant, and these changes are independent of the coordinate system.

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

Path Independence of Potential Energy

The change in potential energy due to a conservative force 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 gravitational potential energy is the same for both paths.

• Equation:

Conservation of Mechanical Energy

In a system where only conservative forces act, the total mechanical energy (sum of kinetic and potential energy) is conserved.

• Net work by conservative forces:

• Change in kinetic energy:

• Conservation equation:

• Total mechanical energy:

• In a conservative system:

Additional info: These principles form the basis for analyzing mechanical systems, such as pendulums, roller coasters, and objects in free fall, where energy transformations occur between kinetic and potential forms without loss to non-conservative forces like friction.