Task
A car is parked on a sloping road with the handbrake on. The handbrake acts on the rear wheels. What is the minimum coefficient of static friction required to prevent the car from slipping?
![Car on a sloping road](https://pickedshares.com/wp-content/uploads/2020/11/TM1-25-1024x722.jpg)
Solution
The following video is in German language, but English subtitles are available.
![Reaction forces on the wheels](https://pickedshares.com/wp-content/uploads/2020/11/Tm1-25solved-1024x607.jpg)
The coordinate system will be turned by the angle Alpha for the solution.
\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
\[\tag{1} 0={F_{\mathit{Bx}}}-G \sin{\left( \alpha \right) }\]
\[\tag{2} 0=-G \cos{\left( \alpha \right) }+{F_{\mathit{By}}}+{F_A}\]
\[\tag{3} 0=G a \sin{\left( \alpha \right) }-G b \cos{\left( \alpha \right) }+{F_{\mathit{By}}} \left( c+b\right) \]
\[\tag{4} {F_{\mathit{Bx}}}={F_{\mathit{By}}} {µ_0}\]
\[\tag{5} {F_{\mathit{By}}}=\frac{{F_{\mathit{Bx}}}}{{µ_0}}\]
\[\tag{6} 0=G a \sin{\left( \alpha \right) }-G b \cos{\left( \alpha \right) }+\frac{{F_{\mathit{Bx}}} \left( c+b\right) }{{µ_0}}\]
\[\tag{7} {F_{\mathit{Bx}}}=G \sin{\left( \alpha \right) }\]
\[\tag{8} 0=\frac{G\, \left( c+b\right) \sin{\left( \alpha \right) }}{{µ_0}}+G a \sin{\left( \alpha \right) }-G b \cos{\left( \alpha \right) }\]
\[\tag{9} {µ_0}=\frac{\left( c+b\right) \sin{\left( \alpha \right) }}{b \cos{\left( \alpha \right) -}a \sin{\left( \alpha \right) }}\]
It is obvious that the weight does not play a role in the determination of the required coefficient of static friction, i.e. a discussion of the distribution of the weight force over several wheels per axle is superfluous.