Work vs. Energy

Definitions and Distinctions

In physics, work and energy are closely related but distinct concepts. Work (W) describes a process—the transfer of energy via a force acting over a distance. Energy (E) describes a state—the capacity of an object or system to do work.

• Work: The process of energy transfer due to a force causing displacement.

• Energy: The ability of a system to perform work.

Both work and energy share the same SI unit: the joule (J).

• Joule (J):

• Calorie: An alternative unit,

Energy can exist in many forms and, with some limitations, can be transformed from one form to another.

Types of Energy in Mechanics

Kinetic and Potential Energy

In mechanics, energy is classified into two main types:

• Kinetic energy (K): Energy of motion.

• Potential energy (U): Energy of position or configuration.

The total mechanical energy of a system is:

Another important form is thermal energy (), which is energy due to temperature and transferred as heat.

Work: Definition and Calculation

Work Done by a Force

Work is done when a force acts on a moving object. For a constant force, the work done is:

• If is the angle between the force and the displacement :

• If there is no motion, no work is done.

• If the force acts perpendicular to the direction of motion (), then .

Examples of Work Calculation

• Lifting a book: You lift a 200-g book with constant speed by 30 cm. How much work do you do?

• Throwing a rock: You throw a 200-g rock up, it travels 30 cm up, turns around, then travels 30 cm down. How much work does gravity do?

• Swinging a ball: You swing a ball around your head at constant speed in a circle with circumference 3 m. What is the work done on the ball by the 10 N tension force in the string during one revolution?

Sign of Work

• Positive work (): When an external force does work on the system ().

• Negative work (): When work is done by the system on the environment ().

Net Work and Multiple Forces

Sum of Works

The net work is the sum of the works done by all forces acting on a system:

• Alternatively, the net work is the work done by the net force:

Example: Multiple Forces

• Given three forces acting at different angles, calculate the work done by each and the net work.

Work as a Scalar (Dot) Product

Vector Multiplication

Work is the scalar product (dot product) of force and displacement vectors:

• Therefore, for a constant force:

Example: Work by a Vector Force

• Find the work that a force does on an object that moves through a displacement:

• a.

• b.

• c.

Variable Forces and Work

Integration for Variable Force

For a variable force, work is calculated by integrating the force over the displacement:

• In one dimension:

Example: Work by a Variable Force

• A variable force acts as an object moves from to . Find the work done.

Graphically, work corresponds to the area under the curve in an F-x diagram.

Summary Table: Work and Energy Concepts

Key Takeaways

• Work and energy are fundamental concepts in physics, sharing the same units and closely related through processes and states.

• Work is calculated using the dot product of force and displacement, and can be positive or negative depending on the direction of force relative to motion.

• For variable forces, work is found by integrating the force over the path of motion.

• Mechanical energy includes both kinetic and potential energy, and can be changed by work or heat transfer.