Question

Derive Johann Bernoulli's differential equation for the catenary, \(d y / d x=s / a\), as follows: Let the lowest point of the hanging cord be the origin of the coordinate system, and consider a piece of the chord of length \(s\) over the interval into a closed-form expression. \([0, x]\). Let \(T(x)\) be the (vector) tension of the cord at the point \(P=(x, y)\). Let \(\alpha\) be the angle that \(T(x)\) makes with 3. Derive Johann Bernoulli's differential equation for the the horizontal and let \(\rho\) be the density of the cord. Show that the equilibrium of horizontal forces gives the equation. equation \(d s=\frac{\sqrt{a} d y}{\sqrt{x}}\) \(|T(0)|=|T(x)| \cos \alpha\), while that of the vertical forces gives \(\rho s=|T(x)| \sin \alpha .\) Since \(d y / d x=\tan \alpha\), Bernoulli's equaDapro tion can be derived by dividing the second equation by the differences of the parts is equal to the difference of the sums first. of the parts.

Short answer

Answer: The Johann Bernoulli's differential equation for the catenary is given by: \(\frac{ds}{dx} = \frac{\sqrt{a} (\rho s)}{\sqrt{x} |T(0)|}\), where \(s\) is the length of the chord, \(\rho\) is the density of the cord, \(|T(0)|\) is the magnitude of the tension at the origin, and \(a\) is a constant.

Step-by-step solution

Identify the given information and define the variables

We are given the following information: - Lowest point of the hanging cord is the origin of the coordinate system. - Length of the chord, s, is defined over the interval [0, x]. - The vector tension of the cord at the point P=(x,y) is represented by T(x). - The angle α represents the angle between the vector tension T(x) and the horizontal. - The density of the cord is denoted by ρ.

Write the equilibrium of horizontal forces equation

We know that the equilibrium of horizontal forces can be given by the equation: \(|T(0)|=|T(x)|\cos \alpha\) Here, \(|T(0)|\) represents the magnitude of the tension at the origin and \(|T(x)|\) represents the magnitude of the tension at point P.

Write the equilibrium of vertical forces equation

Similarly, we can write the equilibrium of vertical forces as: \(\rho s = |T(x)|\sin \alpha\)

Solve for tan(α) using horizontal and vertical forces equations

We can find the tangent of the angle α by dividing the second equation by the first equation, which is: \(\tan \alpha = \frac{\rho s}{|T(0)|}\) Since we know that \(\frac{dy}{dx} = \tan \alpha\), we can write the equation as: \(\frac{dy}{dx} = \frac{\rho s}{|T(0)|}\)

Obtain the catenary differential equation

We are given the catenary differential equation as: \(ds = \frac{\sqrt{a} dy}{\sqrt{x}}\) Substituting the value of \(\frac{dy}{dx}\) from Step 4 to obtain the final differential equation: \(\frac{ds}{dx} = \frac{\sqrt{a} (\rho s)}{\sqrt{x} |T(0)|}\) The final Johann Bernoulli's differential equation for the catenary is given by: \(\frac{ds}{dx} = \frac{\sqrt{a} (\rho s)}{\sqrt{x} |T(0)|}\)

Key concepts

Differential Equations

Differential equations are mathematical tools used to describe the relationship between varying quantities and their rates of change. These equations play a vital role in engineering, physics, economics, and other sciences, as they help in understanding how different variables interact with each other over time or space.

When we derive Johann Bernoulli's equation for the catenary, we're creating a differential equation that models the shape of a hanging cord under its own weight. Differential equations like this are particularly useful because they can express complex natural phenomena with relatively simple mathematical expressions, which can then be solved to predict behavior and characteristics of the system being studied. For our specific case, the equation takes into account factors like the tension in the cord and its density to set up a relationship between the length of the cord and its displacement.

Catenary Curve

The catenary curve represents the idealized shape that a hanging flexible chain or cable assumes under the influence of gravity when supported only at its ends. This particular curve is not just a theoretical interest; it's seen in the real world in the design of arches, bridges, and the sagging of power lines.

Mathematically, it is described by a hyperbolic cosine function, but Bernoulli's differential equation offers another way to explore its properties. This equation helps in analyzing the curve's behavior and properties such as the relationship between the horizontal displacement and the length of the cord. Visualizing this curve can help students understand complex principles like equilibrium of forces in a tangible way, as the catenary curve naturally balances these forces.

Equilibrium of Forces

When we talk about the equilibrium of forces, we refer to a state in which all the forces acting upon an object are balanced so that the object remains in a constant velocity or stationary. For the catenary, we achieve this equilibrium by balancing the horizontal and vertical forces acting on the cord.

The exercise involves examining the forces at work along the cord and deducing their consequences. Horizontal equilibrium is maintained when the tension at any point in the cord is proportional to the cosine of the angle formed with the horizontal. For vertical equilibrium, the weight of the cord, which relates to its density and length, is proportional to the sine of that angle. Analyzing these equilibrium conditions can give students insight into how force vectors contribute to the stability of structures and systems in engineering and physics.

Vector Tension

Vector tension is essentially the pulling force exerted by a string, cable, or similar object when subjected to a load. It's a vector because tension has both magnitude and direction. In the case of the catenary curve, the vector tension varies along the curve and can be decomposed into vertical and horizontal components.

Understanding vector tension is crucial for solving problems related to structures under tension like suspension bridges or the hanging cable in this exercise. By analyzing vector tension at a point, we can infer important information about the system's response to forces, such as stress distribution, potential deformation, and ultimately, whether or not the system will maintain its integrity. Mastery of vector analysis is a foundational element for anyone pursuing disciplines like civil, mechanical, or structural engineering.