# Chapter 6: Work and Energy

Difficulty Level: At Grade Created by: CK-12

Energy is a common concept in the modern world.  A key to understanding energy in detail is how different kinds of energy transform from one to the other, such as circular motion being changed into electrical power.

Chapter Outline

### Chapter Summary

• Work is defined as \begin{align*}W = F_x x = (F \cos \theta)x\end{align*}, the product of the component of the force along the line of motion and displacement.
• Work has units of N*m, or \begin{align*}(\text{kg} \cdot \text{m}^2 / \text{s}^2)\end{align*}, also known as Joules (J).
• Energy comes in two forms: kinetic and potential

Kinetic energy is the energy of motion and potential energy is the energy of position. Potential energy can also have many forms. For example: gravitational potential energy, elastic potential energy, chemical potential energy, and nuclear potential energy to name a few.

• Energy is the ability to do work and has units of joules, J.
• Kinetic energy has the form: \begin{align*}KE = \frac{1}{2} mv^2\end{align*}
• Gravitational potential energy has the form: \begin{align*}PE = mgh\end{align*}
• The potential energy stored in a spring has the form: \begin{align*}PE = \frac{1}{2} kx^2\end{align*}
• The relationship between the applied force and the distance a spring is stretched or compressed is: \begin{align*}F = kx\end{align*} (Hooke’s Law)
• The Work-Energy principle is \begin{align*}W = \Delta KE\end{align*}
• Dissipative forces such as friction are considered non-conservative forces. Mechanical energy is not conserved in the presence of non-conservative forces.
• The conservation of mechanical energy can be written as: \begin{align*}KE_i + PE_i = KE_f +PE_f\end{align*}
• In the presence of dissipative forces the conservation of energy can we written as:
• \begin{align*}KE_i + PE_i = KE_f + PE_f + Q_f\end{align*}, where \begin{align*}Q_f\end{align*} is the energy that has been transformed into heat.
• In the event of an explosion, heat, \begin{align*}Q_i\end{align*}, is added to the initial \begin{align*}KE\end{align*} and \begin{align*}PE\end{align*} energies and in the most general case, heat can also be lost, \begin{align*}Q_f\end{align*}, after the explosive, thus:
• \begin{align*}KE_i + PE_i + Q_i = KE_f + PE_f + Q_f\end{align*}
• The average power is the rate at which work is done or consumed or produced : \begin{align*}P = \frac{W}{t} = \frac{(Fx)}{t} = Fv\end{align*} and has units of Watts \begin{align*}(W)\end{align*}

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Jun 10, 2014