Solving Equation for Suspended Rope: Fixed Ends, Linear Elasticity

[shaner-baner]

I have some questions about the equation governing a suspended rope. I have looked in diffy-Q books and searches a little bit on google and haven't found a satisfactory answer. specifically I am interested in the system with the following assumptions:
1. rope is fixed at both ends
2. rope is linearly elastic (f=kx type law)
3. rope has linear density lambda

most of the derivations I have seen make simplifying assumtions about the
rope that it doesn't stretch or that tensions are colinear
on a differential segment.

Thanks in advance

 

[gtoguy97]

for any help.The equation governing a suspended rope with the assumptions you have described is a wave equation, which describes the motion of a wave on a rope. This equation is given by: $\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$,where $u$ is the displacement of the rope from its equilibrium position, $t$ is time, and $c$ is the wave speed on the rope, which is determined by the linear density $\lambda$ and linear elasticity $k$ according to the equation: $c = \sqrt{\frac{k}{\lambda}}$. This equation can be used to solve for the displacement of the rope at any point in time and space.

 

[tacman]

Thank you for your question. Solving equations for a suspended rope with fixed ends and linear elasticity is a complex topic and there are many different approaches and assumptions that can be made. I will try my best to provide a general overview and address your specific questions.

Firstly, it is important to understand that the equation governing a suspended rope is a partial differential equation known as the wave equation. This equation describes the motion of a flexible string or rope and is derived from the principles of Newtonian mechanics and Hooke's law.

Now, let's address your assumptions one by one.

1. Rope is fixed at both ends: This means that the ends of the rope are attached to immovable objects, such as walls or poles. This assumption simplifies the analysis as it eliminates the need to consider the motion of the ends of the rope.

2. Rope is linearly elastic: This means that the restoring force in the rope is proportional to the amount of stretch or compression, according to Hooke's law. Mathematically, this can be represented as F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.

3. Rope has linear density lambda: This assumption means that the mass per unit length of the rope is constant throughout its length. This is important in determining the tension in the rope at any given point.

Now, in order to solve the wave equation for a suspended rope, we need to make some additional assumptions. One common approach is to assume that the rope is inextensible, meaning that it does not stretch or compress under tension. This simplifies the analysis and allows us to use the one-dimensional wave equation.

Another important assumption is that the tensions in the rope are colinear, meaning that they act along the length of the rope. This is necessary in order to apply the principle of superposition, which states that the total displacement of a point on the rope is the sum of the individual displacements caused by each tension.

It is worth noting that these assumptions are idealizations and may not accurately represent real-life situations. However, they provide a good starting point for understanding the behavior of a suspended rope.

In conclusion, solving equations for a suspended rope with fixed ends and linear elasticity involves making simplifying assumptions in order to derive the wave equation. These assumptions may not always hold true in real-life scenarios, but they provide a good framework for understanding the behavior of the rope. I hope