> [!summary]
Work is a measurement of the amount of energy transferred when a force moves an object
>
**Key Equations:**
>
General work equation:
$W = \int_r ^{r_f} F(r) \cdot dx$
>
If force is non-constant:
$W = F \cdot \Delta x$
>[!info]+ Read Time
**⏱ 2 mins**
# Definition
Work is a scalar and measures the amount of energy transferred when a force moves an object a distance. Mathematically, it's defined as the integral of [[Forces|force]] [[Dots Product & Angles|dotted]] with displacement.
$
\begin{array}{c}
W = \int_{a}^b \vec{F}\cdot \vec{dx} \\
\\
\text{If force is constant:} \\
W = \vec{F} \cdot \Delta x = \vec{F} \Delta x \cos (\theta)
\end{array}
$
> [!note]
There is a [[Dots Product & Angles|dot product]] between the force and displacement because only the component in the direction of displacement contributes to the work done.
>
Physically, this makes. Imagine pushing a box at an upward angle, the force done upward won't contribute to pushing the box forward (in the direction of displacement)
> [!note]+ Work Diagram
![[wor_1.png]]
[^1]
Only the force in the same direction of displacement contributes to work. The perpendicular force will not contribute to a change in motion in the direct of displacement.
---
<!-- Light Mode Newsletter Embed -->
<div class="mm-form-light">
<iframe
src="https://updates.cyberleadhub.com/widget/form/Y0kpQVpjJQuxEfX59m17"
id="inline-Y0kpQVpjJQuxEfX59m17"
title="Join Math & Matter Newsletter (Light)"
data-height="900"
scrolling="no"
allowtransparency="true"
loading="lazy"
style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden"
></iframe>
</div>
<!-- Dark Mode Newsletter Embed -->
<div class="mm-form-dark">
<iframe
src="https://updates.cyberleadhub.com/widget/form/lbeDLm24VjuaFxhjccA1"
id="inline-lbeDLm24VjuaFxhjccA1"
title="Join Math & Matter Newsletter (Dark)"
data-height="900"
scrolling="no"
allowtransparency="true"
loading="lazy"
style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden"
></iframe>
</div>
<!-- Provider script (only once) -->
<script src="https://updates.cyberleadhub.com/js/form_embed.js"></script>
[^1]: Taken from https://tikz.net/energy_work/ by Izaak Neutelings (September 2020)