Chapter 7: Work and Energy

Mathematical Framework

Physics describes the motion of objects using Newton's three laws of motion, with force as a central concept. The calculation and addition of forces utilize a vector framework, meaning forces have both magnitude and direction. This framework allows us to analyze the motion and interactions of objects in space. In this chapter, we introduce the concept of energy and develop new methods to calculate kinematic quantities.

• Force: A push or pull acting on an object, described by Newton's laws.

• Vector Framework: Forces are vectors, so their effects depend on direction as well as magnitude.

• Energy: The ability to do work or cause change; a scalar quantity.

Definition of Work

Work is defined as a force acting over a distance. When a force causes an object to be displaced, work is done on that object. The amount of work depends on both the magnitude of the force and the displacement, as well as the angle between them.

• Formula for Work: where is the angle between the force and displacement vectors.

• Units: The SI unit of work is the joule (J), where .

• Dot Product: The dot product in the formula ensures only the component of force in the direction of displacement contributes to work.

• Maximum Work: Occurs when force and displacement are aligned ().

Example: Work Done on a Box

Suppose you push a box across the floor by applying a force. You can calculate its acceleration using and kinematics, or you can find the work done on the box using the work formula. If the force is not aligned with the displacement, only the component in the direction of motion does work.

• Given: Force applied at angle to displacement .

• Work Done:

Additional info: The notes mention the importance of the dot product and vector components, which is foundational for understanding work in physics.