Problem 70
Question
Suspension bridge cables hang in parabolas The suspension bridge cable shown in the accompanying figure supports a uniform load of \(w\) pounds per horizontal foot. It can be shown that if \(H\) is the horizontal tension of the cable at the origin, then the curve of the cable satisfies the equation $$ \frac{d y}{d x}=\frac{w}{H} x $$ Show that the cable hangs in a parabola by solving this differential equation subject to the initial condition that \(y=0\) when \(x=0\) .
Step-by-Step Solution
Verified Answer
The cable hangs in a parabola: \( y = \frac{w}{H} \cdot \frac{x^2}{2} \).
1Step 1: Identify the Differential Equation
The given differential equation is \( \frac{d y}{d x} = \frac{w}{H} x \), where \( w \) is the load per horizontal foot and \( H \) is the horizontal tension at the origin.
2Step 2: Integrate the Differential Equation
To find \( y \), we integrate the equation \( \frac{d y}{d x} = \frac{w}{H} x \) with respect to \( x \). This yields \( y = \int \frac{w}{H} x \, dx \).
3Step 3: Perform Integration
Integrating \( \frac{w}{H} x \) gives \( y = \frac{w}{H} \cdot \frac{x^2}{2} + C \), where \( C \) is the constant of integration.
4Step 4: Apply Initial Condition
The initial condition given is \( y = 0 \) when \( x = 0 \). Substitute these values into \( y = \frac{w}{H} \cdot \frac{x^2}{2} + C \) to find \( C \).
5Step 5: Solve for the Constant of Integration
Substituting \( x = 0 \) and \( y = 0 \) gives \( 0 = \frac{w}{H} \cdot \frac{0^2}{2} + C \), which simplifies to \( C = 0 \).
6Step 6: Write the Final Equation of the Parabola
With \( C = 0 \), the equation simplifies to \( y = \frac{w}{H} \cdot \frac{x^2}{2} \). This is the equation of a parabola, verifying that the cable hangs in a parabolic shape.
Key Concepts
CalculusSuspension BridgesParabolaIntegrationTension
Calculus
Calculus is a branch of mathematics focusing on change and motion. Two primary concepts in calculus are differentiation and integration.
Differentiation involves finding the derivative, a measure of how a function changes as its input changes. For instance, in our original problem, we begin with the derivative \( \frac{d y}{d x} = \frac{w}{H} x \). This equation relates the change in height \( y \) of the cable to its horizontal distance \( x \), weighted by the constants \( w \) and \( H \).
Integration, on the other hand, is the reverse process, finding a function given its rate of change. By integrating, we find the equation that describes the entire curve of the cable over its domain. Together, these tools of calculus help us model and solve complex real-world problems like the shape of a suspension bridge cable.
Differentiation involves finding the derivative, a measure of how a function changes as its input changes. For instance, in our original problem, we begin with the derivative \( \frac{d y}{d x} = \frac{w}{H} x \). This equation relates the change in height \( y \) of the cable to its horizontal distance \( x \), weighted by the constants \( w \) and \( H \).
Integration, on the other hand, is the reverse process, finding a function given its rate of change. By integrating, we find the equation that describes the entire curve of the cable over its domain. Together, these tools of calculus help us model and solve complex real-world problems like the shape of a suspension bridge cable.
Suspension Bridges
Suspension bridges are a type of bridge where the deck (the load-bearing portion) is hung below suspension cables on vertical suspenders. The cables take the form of parabolas due to the uniform load distributed across the bridge by gravity.
The horizontal tension \( H \) in the cable at the bridge's midpoint helps support the load. It's key to ensure the entire structure remains stable and safe under load. The differential equation \( \frac{d y}{d x} = \frac{w}{H} x \) gives insight into how the properties of the load per horizontal foot \( w \) and tension \( H \) directly affect the parabola's shape and the bridge's integrity.
The horizontal tension \( H \) in the cable at the bridge's midpoint helps support the load. It's key to ensure the entire structure remains stable and safe under load. The differential equation \( \frac{d y}{d x} = \frac{w}{H} x \) gives insight into how the properties of the load per horizontal foot \( w \) and tension \( H \) directly affect the parabola's shape and the bridge's integrity.
Parabola
A parabola is a U-shaped curve that can open up or down. It is characterized by the property that any point on it is equidistant from a fixed point (focus) and a line (directrix).
In the context of suspension bridges, the shape of the cable conforms to a parabolic curve as we've verified with the equation \( y = \frac{w}{H} \cdot \frac{x^2}{2} \). This mathematical concept ensures the bridge can distribute forces evenly, supporting loads effectively.
The specific equation shows how the cable's shape depends on factors such as the load \( w \) and tension \( H \). Recognizing this parabolic nature is crucial in engineering, as it informs design and construction principles.
In the context of suspension bridges, the shape of the cable conforms to a parabolic curve as we've verified with the equation \( y = \frac{w}{H} \cdot \frac{x^2}{2} \). This mathematical concept ensures the bridge can distribute forces evenly, supporting loads effectively.
The specific equation shows how the cable's shape depends on factors such as the load \( w \) and tension \( H \). Recognizing this parabolic nature is crucial in engineering, as it informs design and construction principles.
Integration
Integration is a fundamental concept in calculus, allowing us to determine the original function from its derivative. This is essential when modeling physical phenomena, such as the curve of a suspension bridge cable.
In our problem, integrating \( \frac{d y}{d x} = \frac{w}{H} x \) with respect to \( x \) provides the function \( y = \frac{w}{H} \cdot \frac{x^2}{2} + C \), where \( C \) is the constant of integration. Applying the initial condition \( y = 0 \) when \( x = 0 \), we find \( C = 0 \).
This process illustrates how integration can help determine an object's position or state given its rate of change, a versatile tool in many fields beyond engineering, including physics and economics.
In our problem, integrating \( \frac{d y}{d x} = \frac{w}{H} x \) with respect to \( x \) provides the function \( y = \frac{w}{H} \cdot \frac{x^2}{2} + C \), where \( C \) is the constant of integration. Applying the initial condition \( y = 0 \) when \( x = 0 \), we find \( C = 0 \).
This process illustrates how integration can help determine an object's position or state given its rate of change, a versatile tool in many fields beyond engineering, including physics and economics.
Tension
Tension refers to the force conducted along a cable or other flexible connector, such as the main cables of a suspension bridge. In our problem, the horizontal tension \( H \) plays a significant role.
It ensures that the cable maintains its structural integrity while under a constant load. The tension counteracts the load \( w \) applied per horizontal foot.
Understanding the interplay between tension and load is critical for the design of suspension bridges. If the tension is too low, the bridge could sag dangerously. If too high, the cable could snap. Thus, the tension must be optimized to ensure both safety and functionality.
It ensures that the cable maintains its structural integrity while under a constant load. The tension counteracts the load \( w \) applied per horizontal foot.
Understanding the interplay between tension and load is critical for the design of suspension bridges. If the tension is too low, the bridge could sag dangerously. If too high, the cable could snap. Thus, the tension must be optimized to ensure both safety and functionality.
Other exercises in this chapter
Problem 69
If lines are drawn parallel to the coordinate axes through a point \(P\) on the parabola \(y^{2}=k x, k>0,\) the parabola partitions the rectangular region boun
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