As part of a school mathematics investigation, I am considering how the shape of a catenary would change if the acceleration at each point is produced by centripetal force rather than gravitational force. While in the standard catenary the magnitude of potential energy of a rope at a certain point x is directly proportional to the height at that point, with a rotating rope/chain, the potential energy is proportional to the radius (height) squared.
In the final step of this working out, I have come across the following situation:
$$ F(y) = \int{f(y)dy}\tag{Nonelementary Integral}\\ $$ $$\int_{x_1}^{x_2}{\sqrt{1 + F(x)^2}dx} = l \tag{1}$$ $F(y) = x + c_2 \tag{2}$
By substituting points $(x_1, y_1)$ and $(x_2, y_2)$, the following equations are obtained. $F(y_1) = x_1 + c_2\tag{3}$ $F(y_2) = x_2 + c_2 \tag{4} $
$f(y)$ contains unknowns of $c_1$ and $\lambda$, meaning that the three unknowns $c_1, c_2$ and $\lambda$ must be solved using (3), (4) and (1).
$F(x)$ in (1) is (2) rearranged for $y$.
By finding all of the unknowns, $F(x)$ will express the shape of the rotating caternary-ish shape.
For reference, the points $(x_1, _1)$ and $(x_2, y_2)$ represent locations of the two ends of the rope, and $l$ represents the length of the rope (arc length).
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I'm sure I haven't represented the maths in an ideal way, but I hope it makes sense. It really comes down to, how do I solve this system of equations when the integral is non-elementary? I presume some kind of numeric method, but I would appreciate if I could have any pointers to how I might approach in doing this.
Thanks in advance.
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Due to my limited my limited mathematical skill, I am not able to describe the problem efficiently. It is somewhat messy, but I will describe the details of the problem here.
$$\int{\frac{1}{\sqrt{\frac{\frac{my_1^2}{2} -my_1y + \frac{m}{2}y^2 + \lambda}{c_1}-1}}dy} = x + c_2$$
I start with the equation above, containing unknown constants $c_1$, $c_2$, $\lambda$, and known constant $m$. In the final shape of the rope that I am trying to find, there are two known points $(x_1, y_1)$ and $(x_2, y_2)$ which represent the start and end point of the rope respectively.
If you were to solve the integral and rearrange for y, then that function would draw out the actual shape of the rope. However, this cannot be done until the three unknowns are found. The three unknowns will be found using three equations:
Equation 1: is found by directly substituting $(x_1, y_1)$ into the equation.
Equation 2: is found by substituting $(x_2, y_2)$ into the equation.
Equation 3: is found by identifying $\frac{dy}{dx}$ of the equation, and substituting that into $$\int_{x_1}^{x_2}{\sqrt{1 + \frac{dy}{dx}} dx} = l$$. In the standard catenary derivation, $\frac{dy}{dx}$ is found by rearranging the solved integral into $y$ in terms of $x$, and then finding the derivative.
Now there are three equations with some known constants and three unknowns, which should be able to be solved.