Work and Kinetic Energy: Chapter 7 Study Notes

Work and Kinetic Energy

Introduction

This chapter introduces the concepts of work and kinetic energy, fundamental ideas in classical mechanics. Understanding how forces transfer energy to objects through work, and how this relates to kinetic energy, is essential for analyzing physical systems.

Forms of Energy

Types of Energy

Energy exists in various forms, each with distinct characteristics and applications in physics:

• Mechanical Energy: The sum of kinetic and potential energy in a system.

  • Kinetic Energy: Associated with motion.

  • Potential Energy: Associated with position or configuration.

  Focus for now: Mechanical energy

• Chemical Energy: Stored in chemical bonds, released during chemical reactions.

• Electromagnetic Energy: Energy carried by electromagnetic waves (e.g., light, radio).

• Nuclear Energy: Energy stored in the nucleus of atoms, released in nuclear reactions.

Example: A moving car possesses mechanical energy (kinetic and potential), while a battery stores chemical energy.

Work

Definition and Formula

Work is the process of energy transfer to or from an object via the application of force along a displacement. It is a scalar quantity and provides a direct link between force and energy.

• Formula:

$W = (F \cos \theta) \Delta x$

• F: Magnitude of the applied force

• Δx: Magnitude of the object's displacement

• θ: Angle between the force and displacement vectors

• Work is a scalar quantity

Units:

• SI Unit: Newton-meter (Joule, J)

• US Customary Unit: foot-pound (ft·lb)

• $1 \text{ J} = 1 \text{ kg} \cdot \text{m}^2 / \text{s}^2$

Properties of Work

• Work is zero if the force is perpendicular to the displacement ($\cos 90^\circ = 0$).

• If multiple forces act on an object, the total work is the algebraic sum of the work done by each force.

• Work can be positive (force and displacement in the same direction) or negative (force and displacement in opposite directions).

Example: Lifting a box vertically involves positive work by the lifting force; lowering it involves negative work.

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy

Kinetic energy is the energy of motion. For an object of mass m moving at velocity v:

$K = \frac{1}{2} m v^2$

• Kinetic energy increases with mass and the square of velocity.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:

$W_{net} = \Delta K = K_f - K_i = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$

• Positive net work increases speed (and kinetic energy).

• Negative net work decreases speed (and kinetic energy).

Example: A hammer striking a nail does work, transferring its kinetic energy to the nail.

Work Done by a Variable Force: The Spring

Hooke's Law and Spring Force

The force exerted by a spring is proportional to its displacement from equilibrium, described by Hooke's Law:

$F = -k x$

• k: Spring constant (N/m)

• x: Displacement from equilibrium

Work Done by a Spring

For a variable force such as a spring, work is calculated using integration:

$W = \int F(x) dx$

For a spring:

$W = \frac{1}{2} k x^2$

• Work done by the spring is stored as potential energy.

Example: Compressing a spring stores energy, which can be released to do work.

Power

Definition and Units

Power is the rate at which work is done or energy is transferred:

$P = \frac{dW}{dt}$

• SI Unit: Watt (W)

• US Customary Unit: Horsepower (hp)

• $1 \text{ hp} = 746 \text{ W}$

Electric bills often use the kilowatt-hour (kWh) as a unit, representing energy consumption over time.

Summary Table: Forms of Energy

Type of Energy	Description	Example
Mechanical	Energy due to motion or position	Moving car, stretched spring
Chemical	Energy stored in chemical bonds	Battery, food
Electromagnetic	Energy carried by electromagnetic waves	Light, radio waves
Nuclear	Energy stored in atomic nuclei	Nuclear reactor, sun