Question

A uniform flexible chain of given length is suspended at given points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right) .\) Find the curve in which it hangs. Hint: It will hang so that its center of gravity is as low as possible.

Short answer

The chain hangs in a catenary curve: \ y = a \, \cosh \left( \frac{x}{a} + b \right) + c \ where a, b, and c are determined by the points \( x_1, y_1 \) and \( x_2, y_2 \).

Step-by-step solution

Understand the Problem

A uniform chain suspended between two points will form a curve called a catenary. This curve minimizes the potential energy of the chain making its center of gravity as low as possible.

Define the Catenary Equation

The general equation of a catenary hanging between two points \( x_1, y_1 \) and \( x_2, y_2 \) is given by: \ y = a \, \cosh \left( \frac{x}{a} + b \right) + c \

Determine Constants

Identify the constants a, b, and c by using the boundary conditions provided by points \( x_1, y_1 \) and \( x_2, y_2 \). Solve the system of equations generated by substituting the points into the catenary equation.

Apply Boundary Conditions

Substitute \( x_1, y_1 \) and \( x_2, y_2 \) into \ y = a \, \cosh \left( \frac{x}{a} + b \right) + c \ to form a system of equations to solve for a, b, and c.

Solve the System

Solve the simultaneous equations for a, b, and c using algebraic methods or numerical solutions, which can be performed using software tools like MATLAB or Mathematica.

Write the Final Equation

After determining the values of a, b, and c, write the specific catenary equation for the chain suspended between \( x_1, y_1 \) and \( x_2, y_2 \). This is the curve in which the chain will hang.

Key concepts

catenary equation

When a flexible chain or cable hangs between two points, it naturally forms a curve called a catenary. This is because the catenary shape minimizes potential energy. The catenary equation is a specific mathematical formula used to describe this curve. For a chain that hangs between points \( (x_1, y_1) \) and \( (x_2, y_2) \), the catenary equation is:

\[ y = a \, \text{cosh} \left( \frac{x}{a} + b \right) + c \]

Here, \( \text{cosh} \) is the hyperbolic cosine function, and \( a, b, \text{ and } c \) are constants that need to be determined. This equation tells you exactly how the chain will hang based on the specific points it suspends from.

potential energy

The concept of potential energy plays a crucial role in understanding the catenary problem. Potential energy is the energy possessed by an object due to its position relative to other objects. For a chain hanging under the force of gravity, its potential energy is minimized when it hangs in the shape of a catenary.

In simple terms, when the chain adopts the catenary shape, it is at its lowest possible center of gravity. This is the most stable and natural state for the chain, as the force of gravity is evenly distributed along the curve. Thus, finding the catenary shape is not just a geometric problem but also a physical one related to minimizing energy.

boundary conditions

To solve the catenary problem and find the exact curve on which a chain hangs, we need to apply boundary conditions. Boundary conditions are specific conditions that the solution must satisfy. They are based on the given points where the chain is suspended: \( (x_1, y_1) \) and \( (x_2, y_2) \).

By substituting these points into the catenary equation:

\[ y = a \, \text{cosh} \left( \frac{x}{a} + b \right) + c \]

we form a system of equations. Solving this system allows us to find the values of the constants \ (a, b, \text{ and } c) \. This makes the equation specific to the given problem.

Solving these equations can be done either through algebraic methods or numerical methods using software tools like MATLAB or Mathematica. Once solved, we get the precise description of the catenary curve for the chain hanging between the given points.