Solved Problem on Dynamics
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A car with mass m passes through a speed bump, represented by a circumference arc with a radius equal to R, with constant speed. Assuming the acceleration due to gravity equal to g, determine:
a) The reaction force of the road on the car at the highest point of the speed bump;
b) The maximum speed that the car can have at the highest point of the speed bump without the wheels losing contact with the road.
Problem data:
- Mass of car: m;
- Radius of the speed bump: R;
- Acceleration due to gravity: g.
a) We choose a frame of reference pointing upward, in the opposite direction of acceleration due to
gravity.
Drawing a free-body diagram, we have the forces that act on the car, and we apply Newton's Second Law for a circular motion.
Drawing a free-body diagram, we have the forces that act on the car, and we apply Newton's Second Law for a circular motion.
\[
\begin{gather}
\bbox[#99CCFF,10px]
{{\vec{F}}_{cp}=m{\vec{a}}_{cp}} \tag{I}
\end{gather}
\]
Car:
- \( {\vec P}_{g} \): gravitational force;
- \( \vec{N} \): normal reaction force of the road on the car.
\[
\begin{gather}
N-F_{g}=ma_{cp} \tag{II}
\end{gather}
\]
the gravitational force is given by
\[
\begin{gather}
\bbox[#99CCFF,10px]
{F_{g}=mg} \tag{III}
\end{gather}
\]
the centripetal acceleration is given by
\[
\begin{gather}
\bbox[#99CCFF,10px]
{a_{cp}=\frac{v^{2}}{r}} \tag{IV}
\end{gather}
\]
substituting expressions (III) and (IV) into expression (II)
\[
\begin{gather}
N-mg=m\frac{v^{2}}{R}\\[5pt]
N=mg-m\frac{v^{2}}{R}
\end{gather}
\]
factoring the term mg on the right-hand side of the equation
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{N=mg\left(1-\frac{v^{2}}{Rg}\right)}
\end{gather}
\]
b) At the moment the car loses contact with the speed bump, the normal reaction force becomes zero, assuming the condition N = 0 in the solution of the previous item
\[
\begin{gather}
mg\left(1-\frac{v_{max}^{2}}{Rg}\right)=0\\[5pt]
mg-m\frac{v_{max}^{2}}{R}=0\\[5pt]
\cancel{m}g=\cancel{m}\frac{v_{max}^{2}}{R}\\[5pt]
g=\frac{v_{max}^{2}}{R}\\[5pt]
v_{max}^{2}=Rg
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{v_{max}=\sqrt{Rg\;}}
\end{gather}
\]
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .