Problem 83 - Thomas Calculus

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 \(H_{H}\) 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 $$y=\frac{H}{w} \cosh \frac{w}{H} x$$ 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 $$\tan \phi=\frac{d y}{d x}=\sinh \frac{w}{H} x$$ 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

a. \( \frac{dy}{dx} = \sinh \frac{w}{H} x \). b. Tension \( T = w y \).

Step 1: Set Up Problem with Given Definitions

In the given problem, we consider a cable following the hyperbolic curve defined by the function \( y = \frac{H}{w} \cosh \frac{w}{H} x \). This function indicates the height \( y \) of the cable at any horizontal position \( x \). We need to use this information to establish the slope condition and tension on the cable.

Step 2: Calculate Slope at Point P

To find the slope, determine the derivative of \( y \) with respect to \( x \) (\( \frac{dy}{dx} \)). Given \( y = \frac{H}{w} \cosh \frac{w}{H} x \), we differentiate:\[\frac{dy}{dx} = \frac{H}{w} \times \frac{w}{H} \sinh \frac{w}{H} x = \sinh \frac{w}{H} x\]Thus, the slope \( \tan \phi = \sinh \frac{w}{H} x \), which matches the condition given in part (a).

Step 3: Analyze Horizontal Tension Condition

Now, consider the horizontal tension at any point \( P \). For stability (no movement), the horizontal component of tension must be constant and equal to \( H \). The vertical component of tension results from the weight and is \( w y \) (the weight per unit length times the height \( y \)).

Step 4: Apply Trigonometry to Tension Vector

In vector terms, let \( T \) be the total tension at \( P \). The horizontal component is \( H \), and the vertical component, due to the weight, is \( w y \). The angle \( \phi \) satisfies \( \tan \phi = \frac{w y}{H} \), linking the slope condition to tension.

Step 5: Establish Relation Between Tension and Weight

Using the trigonometric relation \( \tan \phi = \frac{wy}{H} = \sinh \frac{w}{H} x \), we know that the total tension \( T = \sqrt{H^2 + (w y)^2} \) should reflect the forces. Simplifying gives us \( T = w y \), showing that tension equals the weight of the cable, confirming that \( T = w y \) at point \( P(x, y) \).

Key Concepts

Hyperbolic Function

The shape of a cable hanging freely between two supports often takes the form of a curve called a catenary or chain curve. The mathematical representation of this curve is a hyperbolic function. For a hanging cable with horizontal tension, the curve follows the equation \(y = \frac{H}{w} \cosh \frac{w}{H} x \). Here, \(H\) represents the horizontal tension at the cable's lowest point, while \(w\) is the weight of the cable per unit length.

Hyperbolic functions like \(\cosh\) and \(\sinh\) behave similarly to the more familiar trigonometric functions \(\cos\) and \(\sin\), but with key differences. Hyperbolic cosine \(\cosh(x)\) is defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\), giving it the shape suited for a hanging cable. Hyperbolic functions model the natural behavior of hanging cables due to gravity's effect. They ensure that the shape balances the forces acting on the cable at every point along its length.

Important points about hyperbolic functions include:

• They are defined using exponential functions.
• They help describe real-world phenomena, such as the shape of arches and bridges.
• They naturally occur in scenarios involving energy minimization, like a hanging cable.

Tension in Cables

Understanding tension in cables is crucial when studying structures where cables are used for support, such as in suspension bridges or power lines. At the lowest point of a hanging cable, the tension is purely horizontal, given as \(H\). As you move along the cable, tension increases due to the weight of the cable itself.

According to the given solution, the total tension \(T\) at any point \(P(x, y)\) on the cable is due to both the horizontal component \(H\) and the vertical component, which is the weight \(w y\). This means the total tension at any point is exactly equal to this component: \(T = w y\).

Key points about tension in cables include:

• The vertical component of tension results from the weight \(w y\).
• The horizontal tension stays constant at \(H\) throughout the cable.
• In equilibrium scenarios, like a hanging cable, the relationship \(T = w y\) shows that tension reflects the weight of cable above the point.

Differentiation

Differentiation is a mathematical process used to find the rate at which a function changes at any point, often described as the slope of a curve at that point. In the context of a hanging cable, differentiation helps determine how steep the cable is at any point \(P(x, y)\).

For our problem, the function \(y = \frac{H}{w} \cosh \frac{w}{H} x\) represents the hyperbolic curve of the cable. To find the slope, we need to differentiate this function with respect to \(x\). Doing so gives \(\frac{dy}{dx} = \sinh \frac{w}{H} x\), which is also equal to \(\tan \phi\), the slope at point \(P\).

Differentiation is crucial in:

• Determining the slope of the cable, which helps understand angles and tension distribution.
• Establishing relationships between different elements of a structure like forces and pressures.
• Helping solve physical and engineering problems by modeling real-world scenarios.