> [!summary]
The center of mass in terms of dynamics describes the force the center of mass will feel and its effect on motion.
>
**Key Equations:**
$F_{ext}=a_{cm}M$
>[!info]+ Read Time
**⏱ 1 min**
# Definition
The center of mass in terms of dynamics describes the [[External & Internal Forces|external forces]] as if all the mass were centred on a point. This describes the motion of the center of mass using [[Newton Laws|Newton's laws]].
## Derivation
> [!warning] Assumptions
To derive an equation to describe the external forces on the center of mass, assume the following:
> - The [[Center of Mass Velocity & Momentum|velocity center of a mass]] in terms of [[Linear Momentum|momentum]] is $v_{cm} = \frac{p_{tot}}{M}\Rightarrow p_{tot}=v_{cm}M$
> - The [[Derivative|derivative]] of [[Velocity|velocity]] is [[Acceleration|acceleration]]
> - The [[Derivative|derivative]] of [[Linear Momentum|momentum]] is the [[Newton Laws|external force]]
$
\begin{array}{c}
\frac{d}{dt} \left[ p_{tot}= v_{cm}M \right] \\
\frac{dp}{dt}=\frac{dv}{dt}M \\
F_{ext}=a_{cm}M
\end{array}
$
# Resources
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