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\begin{document}

\frame{\frametitle{\b Newton's Laws with Friction - 1} \bigskip

{\bf Car on speedway (1):} 
$$ a_x = \frac{v^2}{r}\,,~~a_x\,\cos(\theta) = g\,\sin(\theta)\,.$$
% \bigskip

{\bf Frictional force (2):}  
$$ \mu_k\,g = a\,,~~a=\frac{v}{t}\,,~~d=\frac{a}{2}\,t^2\,.$$
% \bigskip

{\bf Inclined plane (4).} Force due to friction:
$$ F_{\mu} = \mu\,W\,\cos(\theta)\,. $$

{\bf Up an inclined plane with friction (5).}

Distance:
$$ v=v_0-a\,t\,,~~a=g\,\sin(\theta)+\mu_k\,g\,\cos(\theta)\,,~~d=\frac{v_0}{2}\,t\,. $$
Angle for minimized distance:
$$ 0=\frac{d~}{d\theta}\,\frac{const}{f(\theta)} = -\frac{const}{f^2}\, 
   \frac{df}{d\theta}\,. $$

\vskip 20pt }

\frame{\frametitle{\b Newton's Laws with Friction - 2} \bigskip

{\bf Moving an Iceblock (6).} % \bigskip

Maximum speed:
$$ F^{\min}_x = F^{\min}\,\cos(\theta)\,,~~
   F^{\min}_y =F^{\min}\,sin(\theta)\,$$
$$ F^{\min}_x=F^{\min}\,\cos(\theta)=\mu_s\,(~~?~~)\,.$$
Minimization then similar as in previous problem.
\bigskip

{\bf Gravity Pulling Blocks (7):} 
$$ M_2\,(g-a) = M_1\,(~~?~~)\,.     $$
\vskip 20pt }

\end{document}
