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. Traditionally, forces are calculated and added using a vector framework, which accounts for direction and magnitude. In this chapter, we introduce the concept of energy and develop a new method to analyze kinematics using energy principles.

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

• Vector Framework: Forces are vectors, meaning they have both magnitude and direction.

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

• New Method: Energy-based calculations can simplify analysis of motion, especially when forces vary or are complex.

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

If you push a box across the floor by applying a force, you can calculate the work done using the above formula. For example, if and the box moves in the direction of the force, the work done is:

Additional info: If the force is not aligned with displacement, use the cosine of the angle between them.

Key Properties of Work

• Work is a Scalar: Unlike force, work has magnitude but no direction.

• Positive Work: Force acts in the direction of displacement, increasing the object's energy.

• Negative Work: Force acts opposite to displacement, decreasing the object's energy (e.g., friction).

• No Work: If force is perpendicular to displacement (), no work is done.

Work Done by Multiple Forces

When multiple forces act on an object, the total work is the sum of the work done by each force. For example, if a box is pushed by an applied force and opposed by friction:

• Applied Force:

• Frictional Force:

• Net Work:

Example Table: Work by Different Forces

Work Done by a Varying Force

If the force acting on an object changes over the distance, the total work is found by integrating the force over the path:

• General Formula:

• Piecewise Forces: For forces that change in steps, sum the work over each segment.

Example: Work Done by a Spring (Hooke's Law)

• Hooke's Law: , where is the spring constant and is the equilibrium position.

• Work to Stretch/Compress:

• Restoring Force: Always acts opposite to displacement.

Kinetic Energy and the Work-Energy Theorem

Kinetic energy is the energy associated with an object's motion. The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy.

• Kinetic Energy:

• Work-Energy Theorem:

Example: Stopping Distance of a Car

If a car of mass is traveling at speed and brakes to a stop, the work done by friction equals the change in kinetic energy:

• Doubling the speed quadruples the stopping distance (since kinetic energy depends on ).

Summary Table: Key Equations

Applications

• Mechanical Systems: Calculating work and energy helps analyze machines, vehicles, and physical processes.

• Springs: Used in engineering, physics labs, and everyday devices.

• Friction: Understanding energy loss due to friction is crucial in design and safety.

Additional info: These principles form the foundation for more advanced topics in energy conservation, potential energy, and power.