Problem 426 - Calculus Volume 1

Question

For the following exercises, solve each problem. IT] A chain hangs from two posts four meters apart to form a catenary described by the equation \(y=4 \cosh (x / 4)-3\). Find the total length of the catenary (arc length).

Step-by-Step Solution

Answer

The total length of the catenary is approximately 4.168 meters.

Step 1: Understanding the Catenary Equation

The equation of the catenary provided is \( y = 4 \cosh(x/4) - 3 \). The graph of this equation represents a curve hanging between two points.

Step 2: Determine the Limits for the Arc Length

Since the posts are four meters apart and symmetrical about the origin, we assess the domain of \( x \) from \( -2 \) to \( 2 \). This is determined by setting the endpoints, which are at \( x = -2 \) and \( x = 2 \).

Step 3: Formulate the Arc Length Integral

The arc length \( L \) is given by the formula \( L = \int_{a}^{b} \sqrt{1 + (y')^2} \, dx \). Here, \( a = -2 \) and \( b = 2 \).

Step 4: Differentiate the Catenary Function

Differentiate the function \( y = 4 \cosh(x/4) - 3 \) in respect to \( x \). Using the derivative \( \frac{d}{dx}[\cosh(u)] = \sinh(u) \cdot \frac{du}{dx} \) where \( u = x/4 \), we find \( y' = \sinh(x/4) \).

Step 5: Substitute and Simplify in the Arc Length Integral

Substitute \( y' = \sinh(x/4) \) into the arc length formula: \[ L = \int_{-2}^{2} \sqrt{1 + (\sinh(x/4))^2} \, dx = \int_{-2}^{2} \sqrt{\cosh^2(x/4)} \, dx. \] This simplifies to \( L = \int_{-2}^{2} \cosh(x/4) \, dx \) because \( \sqrt{\cosh^2(x/4)} = \cosh(x/4) \) for all \( x \).

Step 6: Evaluate the Integral

Evaluate the integral \( L = \int_{-2}^{2} \cosh(x/4) \, dx \). The antiderivative of \( \cosh(x/4) \) is \( 4 \sinh(x/4) \). Calculate this from \( -2 \) to \( 2 \): \[ L = 4 \sinh(2/4) - 4 \sinh(-2/4). \]

Step 7: Simplify and Finalize the Result

Since \( \sinh(-x) = -\sinh(x) \), the length simplifies to \[ L = 4 \sinh(1/2) - (-4 \sinh(1/2)) = 8 \sinh(1/2). \] Using a calculator, \( \sinh(0.5) \approx 0.521 \), so the total length is \( 8 \times 0.521 \approx 4.168 \).

Key Concepts

Catenary Equation

A catenary is a mathematical curve that describes the shape of a flexible chain or cable when it is supported at its ends and acted upon only by a uniform gravitational force. This curve is a hyperbolic cosine, represented by the equation:

• For a chain hanging between two points: \[ y = a \, \cosh\left( \frac{x}{a} \right) - c \]where \(a\) and \(c\) are constants that influence the curve's shape and position.
• In our equation, \( y = 4 \, \cosh(x/4) - 3 \), it represents the curve of a chain hanging across two posts that are four meters apart.

In practice, the catenary curve is important in the design of bridges, arches, and electrical cables, ensuring a natural and stable configuration under gravity. Understanding how to derive key characteristics like the arc length from this equation is crucial in applied physics and engineering.

Cosh Function

The hyperbolic cosine function, denoted as \( \cosh(x) \), is a fundamental component in describing catenary curves. Unlike the regular cosine function, the hyperbolic cosine is defined as:

• \[ \cosh(x) = \frac{e^x + e^{-x}}{2} \]
• This function reflects a shape similar to that of the cosine but does not oscillate, positioning instead as the average growth between exponential rise and decay.

The \( \cosh \) function is symmetric and always positive. Its role in the catenary equation helps accurately depict the arc's form. When solving for arc length in such problems, recognizing \( \cosh(x) \) as part of the integral simplifies the calculation significantly, leveraging its mathematical properties to yield real-world dimensions.

Integral Calculus

Integral calculus is the branch of mathematics that deals with the calculation of areas, volumes, and other quantities under curves. To find the arc length of a curve, we use definite integrals. The formula for the arc length \( L \) of a function \( y = f(x) \) over an interval \([a, b]\) is:

• \[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx \]
• This integral accounts for the infinite number of infinitesimally small straight-line segments that make up the curve.

In the catenary problem, the role of integration simplifies when substituting the derivative of the catenary equation into the arc length formula. The simplification \( \sqrt{\cosh^2(x)} = \cosh(x) \) helps compute the length through a more straightforward integration process. This arithmetic form of calculus exemplifies its powerful applications in engineering and physics.

Sinh Function

The hyperbolic sine function, expressed as \( \sinh(x) \), plays a critical part in differentiating and integrating hyperbolic functions. Defined as:

• \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \]
• It describes the difference in growth between exponential functions.

The derivative of the hyperbolic cosine, used in finding the arc length in our problem, is the hyperbolic sine function. Key properties of \( \sinh \) include its odd symmetry, i.e., \( \sinh(-x) = -\sinh(x) \), a useful trait when simplifying calculations:

• In the arc length calculation for the catenary, the derivative \( y' = \sinh(x/4) \) acknowledges changes in the curve's slope.
• When evaluating the integral, knowing sch\( \sinh(-x) = -\sinh(x) \) assists in reducing complexity and computing accurate values.

Through these characteristics, \( \sinh \) tightly interweaves with \( \cosh \) to form a complete calculus framework in solving hyperbolic curve problems.