Potential Energy and Energy Conservation

Learning Outcomes

This chapter introduces the concepts of potential energy and energy conservation in physics, focusing on gravitational and elastic potential energy, the distinction between conservative and nonconservative forces, and the use of energy diagrams.

• Gravitational Potential Energy: Application in vertical motion problems.

• Elastic Potential Energy: Application in systems involving springs.

• Conservative vs. Nonconservative Forces: Understanding their roles in energy conservation.

• Energy Diagrams: Visualizing object motion under conservative forces.

Introduction to Energy Concepts

Energy Storage and Transformation

Energy can be stored in various forms and transformed from one type to another. For example, as a sandhill crane descends, its gravitational potential energy is converted into kinetic energy.

• Energy Storage: Energy is held in a system and can be released or transformed.

• Energy Transformation: Energy changes from one form (e.g., potential) to another (e.g., kinetic).

• Example: A falling object converts gravitational potential energy into kinetic energy as it descends.

Gravitational Potential Energy

Definition and Formula

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

• Key Formula:

• m: Mass of the object

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

• y: Vertical position coordinate

• Interpretation: As an object moves upward, increases and so does ; as it moves downward, decreases.

• Example: A basketball falling towards the ground loses gravitational potential energy, which is converted into kinetic energy, increasing its speed.

Work and Gravitational Potential Energy

The change in gravitational potential energy is directly related to the work done by gravity.

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

• Formula for Change:

• Positive Work: If displacement is downward, gravity does positive work.

• Negative Work: If displacement is upward, gravity does negative work and increases.

• Example: Lifting an object increases its gravitational potential energy.

Conservation of Mechanical Energy

Principle of Conservation

The total mechanical energy of a system is the sum of its kinetic and potential energies. If only conservative forces (like gravity) act, the total mechanical energy remains constant.

• Mechanical Energy:

• Kinetic Energy (): Energy due to motion.

• Potential Energy (): Energy due to position.

• Conservation Law:

• Example: A ball dropped from a height converts potential energy into kinetic energy as it falls, but the total mechanical energy remains constant.

Elastic Potential Energy

Definition and Formula

Elastic potential energy is stored in objects that can be stretched or compressed, such as springs.

• Elastic Object: Returns to its original shape after deformation.

• Key Formula:

• k: Spring constant (measure of stiffness)

• x: Displacement from equilibrium (positive for stretching, negative for compression)

• Example: Stretching a spring stores elastic potential energy, which is released when the spring returns to its original length.

Work Done by a Spring

When a spring is stretched or compressed, it does work on the attached object, transferring energy.

• Positive Work: When the spring returns to equilibrium, it does positive work on the object.

• Graph: The elastic potential energy vs. displacement graph is a parabola, always non-negative.

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

Situations with Both Gravitational and Elastic Forces

Total Potential Energy

When both gravitational and elastic forces are present, the total potential energy is the sum of both contributions.

• Formula:

• Example: A mass attached to a spring and suspended vertically experiences both gravitational and elastic potential energy.

Conservative and Nonconservative Forces

Conservative Forces

Conservative forces allow energy to be converted between kinetic and potential forms without loss. Examples include gravity and spring force.

• Properties:

  • Work depends only on initial and final positions, not the path taken.

  • Work is reversible.

  • Work is zero for closed paths (start and end at same point).

  • Associated with a potential energy function.

• Example: Gravity is a conservative force.

Nonconservative Forces

Nonconservative forces, such as friction, dissipate mechanical energy as heat or other forms, and do not have a potential energy function.

• Properties:

  • Work depends on the path taken.

  • Energy is lost from the system (e.g., as heat).

• Example: Friction in a rolling tire increases the tire's internal energy and temperature.

Conservation of Energy

General Law

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another.

• Formula:

• : Change in kinetic energy

• : Change in potential energy

• : Change in internal energy (e.g., due to friction)

• Example: In a system with friction, some mechanical energy is converted to heat.

Force and Potential Energy in One Dimension

Relationship Between Force and Potential Energy

For conservative forces in one dimension, the force can be derived from the potential energy function as the negative derivative with respect to position.

• Formula:

• Interpretation: Where changes rapidly, the force is large. The force always acts to decrease potential energy.

• Example: For a spring, the force pushes the mass toward equilibrium ().

Force and Potential Energy in Three Dimensions

Gradient and Conservative Forces

In three dimensions, the components of a conservative force are given by the negative partial derivatives of the potential energy function with respect to each coordinate.

• Formula:

• Gradient: The vector sum of the partial derivatives gives the direction and magnitude of the force.

• Example: A hiker on a mountain experiences a force pushing them toward lower elevation (lower potential energy).

Energy Diagrams

Visualizing Potential and Mechanical Energy

An energy diagram plots the potential energy function and the total mechanical energy of a system. It helps visualize how an object moves under the influence of conservative forces.

• Equilibrium Points: Where the slope of is zero, the force is zero (equilibrium).

• Stable Equilibrium: At a minimum of , the force restores the object to equilibrium.

• Unstable Equilibrium: At a maximum of , the force moves the object away from equilibrium.

• Example: A glider attached to a spring oscillates around the stable equilibrium point.

Table: Comparison of Conservative and Nonconservative Forces