- #1
IsakVern
- 8
- 0
- TL;DR Summary
- In the classic "loop the loop" problem where a model car rolls from rest down a ramp from a height h0 to perform a "loop the loop" stunt around a circular track of radius R in the vertical plane, why is the minimal starting height required in order to perform a loop without leaving the track exactly 5/2 of the radius R?
The height can be determined by conservation of energy (ignoring all friction). The mechanical energy when the car is at rest, equals the mechanical energy when the car is in the middle of the loop (at the top of the loop):
\begin{equation}
E_{0} = E_{loop}
\\
mgh_0 = \frac{1}{2}mv^2+mgh_{loop}
\\
gh_0 = \frac{1}{2}v^2+2gR
\end{equation}
In the middle of the loop (at the top of the circular track), the only force acting on the car is gravity, which equals the centripetal force:
\begin{equation}
F = mg = \frac{mv^2}{R}
\\
\rightarrow v^2=gR
\end{equation}
Inserting this into the first equation, we get:
\begin{equation}
gh_0 = \frac{1}{2}gR+2gR
\\
h_0= \frac{5}{2}R
\end{equation}
Note that the height doesn't depend on the magnitude of the gravitational force at all, so no matter what planet you perform this experiment on, the minimal height needed in order for the car to loop the loop is always 2.5 times the radius of the loop. Is this just a algebraic coincidence/necessity, or is there some neat explanation for why this is?
\begin{equation}
E_{0} = E_{loop}
\\
mgh_0 = \frac{1}{2}mv^2+mgh_{loop}
\\
gh_0 = \frac{1}{2}v^2+2gR
\end{equation}
In the middle of the loop (at the top of the circular track), the only force acting on the car is gravity, which equals the centripetal force:
\begin{equation}
F = mg = \frac{mv^2}{R}
\\
\rightarrow v^2=gR
\end{equation}
Inserting this into the first equation, we get:
\begin{equation}
gh_0 = \frac{1}{2}gR+2gR
\\
h_0= \frac{5}{2}R
\end{equation}
Note that the height doesn't depend on the magnitude of the gravitational force at all, so no matter what planet you perform this experiment on, the minimal height needed in order for the car to loop the loop is always 2.5 times the radius of the loop. Is this just a algebraic coincidence/necessity, or is there some neat explanation for why this is?