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# 8.2: Key Concepts

Difficulty Level: At Grade Created by: CK-12
• Work is simply how much energy was transferred from one system to another system. You can always find the work done on an object (or done by an object) by determining how much energy has been transferred into or out of the object through forces. If you graph force vs. distance, the area under the curve is work. (The semantics take some getting used to: if you do work on me, then you have lost energy, and I have gained energy.)
• Work can be computed by multiplying the distance traveled with the component of force that is parallel to that distance
• Energy can be transformed from one kind into the other; if the total energy at the end of the process appears to be less than at the beginning, the “lost” energy has been transferred to another system, often by heat or sound waves.
• Work can be computed by multiplying the distance with the component of force that is parallel to the distance
• Efficiency is equal to the output energy divided by the input energy.

## Math of Force, Energy, and Work

When an object moves in the direction of an applied force, we say that the force does work on the object. Note that the force may be slowing the object down, speeding it up, maintaining its velocity --- any number of things. In all cases, the net work done is given by this formula:

W=Fd=FΔx[1]Work is the dot product of force and displacement.\begin{align*}W = \vec{F}\cdot \vec{d} = \vec{F}\cdot \Delta \vec{x} && \text{[1]Work is the dot product of force and displacement.}\end{align*}

In other words, if an object has traveled a distance d\begin{align*} d \end{align*} under force F\begin{align*} \vec{F} \end{align*}, the work done on it will equal to d\begin{align*} d \end{align*} multiplied by the component of F\begin{align*}\vec{F}\end{align*} along the object's path. Consider the following example of a block moving horizontally with a force applied at some angle:

Here the net work done on the object by the force will be Fdcosθ\begin{align*}F d \cos \theta\end{align*}.

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