Problem 83 - Thomas Calculus in SI Units

Question

Hanging cables Imagine a cable, like a telephone line or TV cable, strung from one support to another and hanging freely. The cable's weight per unit length is a constant \(w\) and the horizontal tension at its lowest point is a vector of length \(H\). If we choose a coordinate system for the plane of the cable in which the \(x\) -axis is horizontal, the force of gravity is straight down, the positive \(y\) -axis points straight up, and the lowest point of the cable lies at the point \(y=H / w\) on the \(y\) -axis (see accompanying figure), then it can be shown that the cable lies along the graph of the hyperbolic cosine Such a curve is sometimes called a chain curve or a catenary, the latter deriving from the Latin catena, meaning "chain." a. Let \(P(x, y)\) denote an arbitrary point on the cable. The next accompanying figure displays the tension at \(P\) as a vector of length (magnitude) \(T\), as well as the tension \(H\) at the lowest point \(A .\) Show that the cable's slope at \(P\) is b. Using the result from part (a) and the fact that the horizontal tension at \(P\) must equal \(H\) (the cable is not moving), show that \(T=w y .\) Hence, the magnitude of the tension at \(P(x, y)\) is exactly equal to the weight of \(y\) units of cable.

Step-by-Step Solution

Answer

The magnitude of tension at any point is given by \( T = wy \), equaling the weight of \( y \) units of cable.

Step 1: Express the Slope of the Cable at Point P

Start by understanding that the slope of the cable at any point \( P(x, y) \) is represented by the derivative \( \frac{dy}{dx} \). For the catenary, the general formula for \( y \) is given by \( y = \frac{H}{w} \cosh\left(\frac{wx}{H}\right) \). Therefore, the derivative and hence the slope \( \frac{dy}{dx} \) at any point \( x \) is \( \sinh \left(\frac{wx}{H}\right) \), but we want it in terms of tension at any point \( P \). By definition, \( \tanh(u) = \frac{\sinh(u)}{\cosh(u)} \), so the slope is \( \tanh\left(\frac{wx}{H}\right) \).

Step 2: Relate Slope to Tension Components

The magnitude of the tension \( T \) at point \( P \) and its components can be analyzed. Recall that tension \( T \) in the cable is comprised of a horizontal component equal to \( H \) and a vertical component \( V \) related to weight. By the relationship of components in trigonometry, \( \tan(\theta) = \frac{V}{H} \), where \( \theta \) is the angle made by tension with horizontal. This tangential relation corresponds to the slope given by \( \tanh(u) = \frac{\sinh(u)}{\cosh(u)} \). Substitute \( V = wy \), this gives \( \tanh(\frac{wx}{H}) = \frac{wy}{H} \).

Step 3: Use Physical Constraint for Horizontal Tension

At any point \( P(x, y) \), the condition for equilibrium states that the horizontal component of tension remains constant, that is \( H = T \cos(\theta) \). Since \( \theta = \tanh^{-1}\left(\frac{wy}{H} \right) \), simplify by noting \( \cos(\theta) = \frac{1}{\cosh(u)} \). Therefore, combining these trigonometric identities gives \( H = T \frac{1}{\cosh(\frac{wx}{H})} \).

Step 4: Conclude Magnitude of Tension at Point P

Using the derived equations and constraints, the entire tension \( T \) is related to the vertical component as \( T = H\cosh\left(\frac{wx}{H}\right) = wy \), under equilibrium and no net motion conditions of the cable. As a result, this proves \( T = wy \), where the entire weight of \( y \) units of cable determines the tension \( T \) at any point \( P \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions, much like their trigonometric counterparts, are essential in describing phenomena involving hanging cables, such as the catenary curve. These functions are derived from exponential functions and serve various applications in physics and engineering. Specifically, for hanging cables, the function we focus on is the hyperbolic cosine function, denoted as \( \cosh(x) \). This function can be expressed as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). The hyperbolic sine function, \( \sinh(x) = \frac{e^x - e^{-x}}{2} \), also plays a critical role, especially when calculating slopes or angles in systems of tension.
Understanding these functions allows us to accurately model and predict the shape and behavior of a hanging cable. These functions are pivotal when deriving formulas for the slope of the catenary curve, where \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \) relates to the cable's slope at any given point. By grasping these basics of hyperbolic functions, a complex scenario like the hanging cable becomes much more approachable.

Catenary Curve

The catenary curve is the beautiful shape that a flexible cable forms under its own weight when suspended between two points. Unlike a parabola, which you might expect, a catenary curve specifically takes the form of a hyperbolic cosine function: \( y = \frac{H}{w} \cosh\left(\frac{wx}{H}\right) \). This equation models the equilibrium shape of the cable, where the forces acting upon it, such as tension and gravity, are perfectly balanced.
In Latin, \'catena\' means \'chain\', an apt word picture as the curve mimics the droop of a chain between two posts. The lowest point of this curve occurs at the point \( \left(0, \frac{H}{w}\right) \), where \( H \) is the horizontal tension at that point.
The catenary curve is more than just a visual marvel; it's an elegant solution to the physics problem of a hanging cable. Engineered designs, from bridges to power lines, often incorporate the math behind this curve to ensure structural integrity and optimal material use.

Tension Components

In the analysis of a hanging cable, understanding the components of tension is key. Tension \( T \) at any point on a cable divides into horizontal and vertical components. The horizontal component remains constant throughout the length of the cable and is denoted as \( H \), while the vertical component depends on the weight of the cable section at point \( P(x, y) \).
This vertical component can be expressed as \( V = wy \), where \( y \) is the vertical height of the cable at point \( P \), and \( w \) is the weight per unit length. In trigonometric terms, these components relate to the angle \( \theta \) that \( T \) makes with the horizontal. We use \( \tan(\theta) = \frac{V}{H} \) to relate these, which simplifies the understanding of physical interactions in the system.
It's important to note the equilibrium state in these tension components, since they stabilize the cable and keep it from moving. Recognizing how tension is distributed helps in solving applied physics problems and in practical applications like designing safe and efficient structures.

Trigonometry in Physics

Trigonometry is a valuable tool in physics for solving problems related to angles and forces. In the context of hanging cables, trigonometric principles apply when determining the slope, tension, and other components of a system.
The slope of the cable at any point, represented by \( \tanh\left(\frac{wx}{H}\right) \), comes from the trigonometric identity for hyperbolic functions. Understanding these relationships allows one to calculate how forces are decomposed into vertical and horizontal components, essential for determining the cable's behavior.
In equilibrium, we utilize the trigonometric equality \( \cos(\theta) = \frac{1}{\cosh(u)} \), where \( u \) is the argument of the hyperbolic functions. Physics heavily borrows from trigonometric functions to describe real-world systems in a coherent and systematic way. Knowledge of these principles not only aids in academic pursuits but also lays the foundation for engineering solutions that influence everyday life.