Work & Energy

Introduction to Energy

Energy is a fundamental concept in physics used to track interactions within a system. It provides a quantitative measure of the ability of a system to perform work or produce change.

• Definition: Energy is the score that tracks all interactions in a system.

• Types: Common forms include kinetic energy, potential energy, thermal energy, etc.

• Application: Used to analyze and predict the behavior of physical systems.

Introduction to Work

Work is the basic way to calculate energy transfer in a system. It keeps track of forces that contribute to motion.

• Definition: Work is the process of energy transfer to or from an object via the application of force along a displacement.

• Formula: The general definition of work is given by the integral:

• Explanation: The integral sums the dot product of force and displacement along a path from point a to b.

• Dot Product: where is the angle between the force and the direction of displacement.

• Scalar Product: Only the component of force in the direction of motion contributes to work.

Example: Pushing a box along a straight path with a constant force.

Calculating Work

Work Done by a Constant Force

When a constant force is applied along a straight path, the work done simplifies to:

• F: Magnitude of the force

• D: Magnitude of displacement

• : Angle between the force and displacement direction

Example: Lifting an object vertically with a force equal to its weight ().

Kinetic Energy

Definition and Formula

Kinetic energy is the energy associated with the motion of an object.

• Formula:

• m: Mass of the object

• v: Speed of the object

Work-Energy Theorem: The net work done on an object is equal to the change in its kinetic energy:

• If an object moves with speed , work has been done on it.

• Kinetic energy is a measure of all work done in the past on an object.

Work Done by a Non-Constant Force: The Spring

Spring Force and Hooke's Law

When dealing with springs, the force is not constant but depends on the displacement from equilibrium.

• Spring Force (): The force exerted by a spring pulling the mass back to equilibrium.

• Equilibrium Length (): The length of the unstretched or uncompressed spring.

• Hooke's Law:

• k: Spring constant (stiffness of the spring), units:

• x: Displacement from equilibrium

• Force always acts opposite to displacement.

• k is experimentally determined.

Example: Stretching a spring by pulling a mass attached to it.

Work Done by a Spring

To find the work done by a spring, we integrate the spring force over the displacement:

Substituting Hooke's Law:

Evaluating the integral:

Or, rearranged:

• If , then

• If the mass slides to , then (spring does work to return to equilibrium)

• Work done by a spring scales as displacement squared, not force times distance.

Additional info: For non-constant forces, the integral must be evaluated. In this class, you may not need to perform the full calculation, but the result for spring work is important for future problems.

Conservation of Energy

Energy Conservation Principle

The principle of conservation of energy states that the total energy in a closed system remains constant, though it may change forms.

• Energy: Tracks all interactions in a system.

• Work-Energy Relation:

• Kinetic energy indicates work was done in the past.

• Kinetic energy is the score of past work.

• If we can predict what interactions could happen in the future, we can use energy conservation to analyze the system.

Example: A block sliding on a frictionless surface, where all work done changes the block's kinetic energy.

Summary Table: Key Formulas