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\subsection{Friction}

Energy is conserved at the atomic level. But
mechanical energy appears not to be conserved in
many everyday processes. This is summarised by
friction.

\subsubsection*{Dry Friction}

When solid objects are in contact, the friction
force is of the form
\[ F = \mu R .\]
Here $\mu$ is the \emph{coefficient of friction},
and it is a dimensionless constant (no units). For
example, the value for aluminium on aluminium is
about $0.3$. It is an empirical property which
depends on the two materials touching.
$R$ is the \emph{reaction force}.
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/a646b0c2834711ec.png}
\end{center}

\subsubsection*{Fluid Drag}

When an object moves through a fluid (liquid or
gas), it experiences \emph{drag}. This is
typically of two different kinds:
\begin{itemize}
    \item Linear drag
        \[ \bf{F} = -\gamma \bf{v} \]
        for some constant $\gamma$. Applies for
        objects moving slowly in a viscous fluid
        (for a spherical object of radius $L$,
        Stokes' formula gives
        \[ \gamma = 6\pi \zeta L \]
        where $L$ is the viscosity of fluid).
    \item Quadratic drag
        \[ \bf{F} = -\gamma |\bf{v}| \bf{v} \]
        for some constant $\gamma$. Applies for
        objects moving in less viscous fluids, for
        example objects falling in air (usually
        have $\gamma \propto L^2$).
\end{itemize}

\begin{note*}
    Mechanical energy isn't conserved for either
    of these forces. (Quadratic drag is easier to
    understand than linear).
\end{note*}

\subsubsection*{An example}

Consider a projectile moving under a constant
gravitational force and experiencing linear drag
(ball in treacle). At time $t = 0$, throw ball
with velocity $\bf{u}$. The equation of motion is
\[ m \dfrac{\bf{v}}{t} = m \bf{g} - \gamma \bf{v} \]
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/92893f1c834911ec.png}
\end{center}
Trick: solve for $\bf{v}$ first, then $\bf{x}$. \\
Introduce an integrating factor:
\[ \dfrac{}{t} (e^{\gamma t / m} \bf{v}) =
e^{\gamma t / m} \bf{g} \]
\[ \implies e^{\gamma t / m} \bf{v} =
\frac{m}{\gamma} e^{\gamma t / m} \bf{g} + \bf{c} \]
where $\bf{c}$ is an integration constant. Now
using $\bf{v} = \bf{u}$ at $t = 0$ implies that
$\bf{c} = \bf{u} - \frac{m}{\gamma} \bf{g}$. So
\[ \dot{\bf{x}} = \bf{v} = \frac{m \bf{g}}{\gamma} +
\left( \bf{u} - \frac{m}{\gamma} \bf{g} \right)
e^{-\gamma t / m} \]
\[ \implies \bf{x} = \frac{m \bf{g} t}{\gamma} -
\frac{m}{\gamma} \left( \bf{u} - \frac{m}{\gamma}
\bf{g} \right) e^{-\gamma t / m} + \bf{b} .\]
(for some other integration constant $\bf{b}$).
Now using $\bf{x} = 0$ at $t = 0$,
\[ \implies \bf{x} = \frac{m \bf{g} t}{\gamma} +
\frac{m}{\gamma} \left( \bf{u} - \frac{m}{\gamma}
gbf\right) (1 - e^{-\gamma t / m}) .\]
In components,
\[ \bf{x} =
\begin{pmatrix}
    x \\
    y \\
    z
\end{pmatrix}
\qquad \bf{u} =
\begin{pmatrix}
    u \cos \theta \\
    0 \\
    u \sin \theta
\end{pmatrix}
\qquad \bf{g} =
\begin{pmatrix}
    0 \\
    0 \\
    -g
\end{pmatrix}
\]
\[ \implies x = \frac{m}{\gamma} u\cos\theta (1 -
e^{-\gamma t / m}) \]
Note that it never gets further than
$\frac{m}{\gamma} u \cos\theta$.
\[ z = -\frac{mgt}{\gamma} + \frac{m}{\gamma}
\left( u \sin\theta + \frac{mg}{\gamma} \right) (1
- e^{-\gamma t / m}) \]
(units of $\gamma$ are mass per unit time).

\newpage
\section{Dimensional Analysis}

Physical quantities have units, or
\emph{dimensions}. In any equation, the units have
to be constant / equal in each term. For most
problems, it's convenient to introduce three
fundamental dimensions
\begin{itemize}
    \item Length $L$
    \item Mass $M$
    \item Time $T$
\end{itemize}
The dimension $[Y]$ of quantity $Y$ should be
expressible in terms of $L$, $M$ and $T$. For
example
\begin{itemize}
    \item $[\text{area}] = L^2$
    \item $[\text{speed}] = LT^{-1}$
    \item $[\text{acceleration}] = LT^{-2}$
    \item $[\text{force}] = MLT^{-2}$ (from $F =
        ma$)
    \item $[\text{energy}] = ML^2T^{-2}$ (from $E
        = \half mv^2$)
\end{itemize}
Similarly constants can have dimensions. For
example
\begin{itemize}
    \item $[G] = M^{-1}L^2T^{-2}$ (from $F = -
        \frac{GMm}{r^2}$)
\end{itemize}
For some problems, one needs even less than three
dimensions, and for other problems more (for
example charge).