Identify each energy exchange as primarily heat or work and determine whether the sign of \(\Delta E\) is positive or negative for the system. a. A rolling billiard ball collides with another billiard ball. The first billiard ball (defined as the system) stops rolling after the collision. b. A book falls to the floor. (The book is the system.) c. A father pushes his daughter on a swing. (The daughter and the swing are the system.)

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In essence, it explains how energy is converted from one form to another and how this affects matter. The fundamental principles of thermodynamics are encapsulated in its four laws.

For instance, the First Law of Thermodynamics, also known as the conservation of energy, states that energy cannot be created or destroyed, only transformed. When looking at the billiard ball collision, the rolling ball comes to a halt, indicating a transformation of its kinetic energy into another form, such as heat or sound, through the work done by the collision.

In the falling book scenario, gravitational potential energy is being converted into kinetic energy as the book accelerates towards the ground. This is a clear illustration of how energy transformation follows the principles of thermodynamics, with no energy lost in the process, only changed from one type to another.

Kinetic energy is the energy possessed by an object due to its motion. It’s directly proportional to the mass of the object and the square of its velocity, described by the equation: \[ KE = \frac{1}{2}mv^2 \]

In the case of the rolling billiard ball, kinetic energy is what allows the ball to move across the table. When the first billiard ball collides with the second, it transfers some of its kinetic energy to the second ball and comes to a stop. This loss of motion signifies a decrease in its kinetic energy, which in terms of physics, could be transformed into other energy forms like sound, heat, or even potential energy in the form of deformation.

The concept of kinetic energy also is applicable in the falling book scenario. As the book falls, its velocity increases, and thus its kinetic energy also increases until the book comes into contact with the floor, at which point the energy is dispersed.

Potential energy, in contrast, is the energy stored within an object due to its position or state. For an object in a gravitational field, we refer to its potential energy associated with its height above a reference point and is given by the formula: \[ PE = mgh \]

where m stands for the mass of the object, g for the gravitational acceleration, and h for the height of the object. When the book is held at a certain height, it possesses gravitational potential energy. As it’s dropped and falls to the floor, that potential energy is converted into kinetic energy. The initial potential energy of the book equals the kinetic energy it gains just before it hits the ground, assuming no energy is lost to air resistance or other forces.

The potential energy concept is less apparent in the swing scenario, but it’s still present. At the highest points in the swing's arc, the daughter has maximum potential energy, which translates into kinetic energy as she swings back down through the lower arc.

The work-energy theorem is a fundamental principle that states the work done on an object is equal to the change in its kinetic energy. Work is done when a force causes displacement in the direction of the force and is calculated as: \[ W = Fd \]

where W is work, F is the force applied, and d is the displacement in the direction of the force. In the context of the three scenarios, we see the work-energy theorem in action in multiple ways. For the billiard balls, the force of the collision does work on the first ball, causing it to stop and its kinetic energy to change. Similarly, when the book falls, the work done by gravity increases the book’s kinetic energy. The father pushing the swing does work on the swing system, which increases its kinetic energy and sometimes its potential energy.

The work-energy theorem is particularly useful because it connects the concept of force (from Newtonian mechanics) with energy (from thermodynamics), bridging the gap between these fundamental physics concepts and illustrating how they are mutually inclusive in describing the motion and interactions of objects.