Question

Identify each energy exchange as primarily heat or work and determine the sign of \(\Delta E\) (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 is dropped 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.)

Short answer

a. Work is performed on the ball, \(\Delta E < 0\). b. Work is performed by gravity, but there is no change in internal energy, \(\Delta E = 0\). c. Work is performed on the daughter and the swing, \(\Delta E > 0\).

Step-by-step solution

Analyzing the Billiard Ball Collision

In the scenario of the colliding billiard balls, energy is transferred through work as one ball exerts a force on the other. The first billiard ball stops, which implies it has lost kinetic energy to the second ball. Since energy is leaving the system (the first ball), the change in energy, \(\Delta E\), for the system is negative.

Determining the Book's Energy Exchange During the Fall

When the book is dropped, it converts potential energy into kinetic energy under the influence of gravity. There is no heat exchange taking place; rather, gravity is doing work on the book. As the book falls, its internal energy is not changing (ignoring air resistance), so \(\Delta E\) for the system is zero.

Understanding Energy Transfer in Pushing the Swing

The father does work on the swing and the daughter by applying a force to push the swing. This adds energy to the system by increasing the kinetic and potential energy of the daughter and the swing. Thus, the change in energy, \(\Delta E\), for the system is positive.

Key concepts

Work-Energy Principle

The work-energy principle is a fundamental concept that bridges the gap between force and motion. At its core, it states that the work done by the forces acting on an object results in a change in kinetic energy of that object. Mathematically, this relationship can be expressed as \( W = \Delta KE \), where \( W \) is the work done and \( \Delta KE \) stands for the change in kinetic energy.

Let's take the example of a rolling billiard ball that stops after colliding with another ball. In this case, the work done on the first ball by the second is ultimately negative, since the motion of the first ball is halted; this translates to a negative change in kinetic energy. Similarly, when a father pushes his daughter on a swing, the work he does on the swing system is positive, increasing the kinetic and potential energies of both the daughter and the swing.

Kinetic and Potential Energy

Kinetic energy (KE) is the energy possessed by an object due to its motion, defined as \( \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object. On the other side, potential energy (PE) is the stored energy of position possessed by an object. For example, gravitational potential energy near Earth's surface is calculated as \( mgh \), with \( m \) being the mass, \( g \) the acceleration due to gravity, and \( h \) the height above a reference point.

Consider a book falling off a table; it starts with potential energy due to its height and as it falls, this potential energy is converted into kinetic energy, increasing its motion speed as it approaches the ground. Throughout this process, as long as we neglect other forces like air resistance, the total mechanical energy (the sum of kinetic and potential energies) remains constant. These concepts show energy transformation within a system and underscore the variety of energy types within physical interactions.

Conservation of Energy

The conservation of energy is a principle stating that the total energy in an isolated system remains constant, provided the system is not subjected to external forces. This law implies that energy can neither be created nor destroyed, but it can be transformed from one form to another or transferred from one object to another.

For instance, when the rolling billiard ball comes to a stop after hitting the second ball, the kinetic energy of the first ball is not lost but transferred to the second ball or transformed into other forms of energy like sound or heat. In the case of the swinging daughter, the energy input by the father isn't just vanishing; it's being stored as potential energy at the height of the swing's arc and converted to kinetic energy as the daughter descends. Understanding this concept aids in solving many physics problems and reaffirming the constancy of energy in our universe.