Chapter 8: Problem 34

A flexible chain of length \(l\) is hung over a peg with one end of the chain slightly longer than the other. Assuming that the chain slides off with no friction, write and solve the differential equation of motion to show that \(y=y_{0} \cosh t \sqrt{2 g / l}, 0

Short Answer

Expert verified

The motion of the chain is described by \( y = y_{0} \cosh \big(t \frac{2g}{l}\big) \).

Step by step solution

Define Variables and Initial Conditions

Let the length of the chain be represented by a variable, and set up the initial conditions of the problem. Define the variable \( y \) to represent the difference in length between the two sides of the chain.

Derive the Differential Equation

Using dynamics and considering that there is no friction, derive the equation of motion for the chain. Taking into account that the system is driven by gravity, set up the differential equation.

Solve the Differential Equation

The differential equation for the motion would be of the form \( \frac{d^2 y}{dt^2} = \frac{2g}{l} y \). This is a second-order linear differential equation.

Find the General Solution

The general solution to the differential equation \( \frac{d^2 y}{dt^2} = \frac{2g}{l} y \) can be found using hyperbolic functions. The general solution to this differential equation is \( y = A \text{cosh} \big(t \frac{\beta}{l} \big) + B \text{sinh} \big(t \frac{\beta}{l} \big) \), where \( \beta = \frac{\beta}{l} \).

Apply Initial Conditions

Using the initial conditions given in the problem, set \( y = y_0 \) when \( t = 0 \). Substitute these values into the general solution to find the constants. This simplifies the solution to match the required form.

Simplify the Final Expression

Substitute the constants back into the equation. The correct form is achieved by noting that \( B = 0 \) and hence the final simplified form is \( y = y_{0} \text{cosh} \big(t \frac{ \beta }{l} \big) \), where \( \beta = \frac{2g}{l} \). Thus the desired solution is confirmed as \( y = y_{0} \text{cosh} \big(t \frac{\beta}{l} \big) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

chain motion

Chain motion refers to the way a flexible chain behaves when it is subjected to forces, in this case, gravity. The problem set describes a chain hanging over a peg with no friction, which means that the dynamics solely depend on gravitational forces pulling it downward.
Understanding the motion of the chain starts with defining variables. Here, we define 'y' as the difference in length between the two sides of the chain.
The initial conditions given in the problem show that at time t=0, the chain has a length difference of y=y_0.

second-order differential equation

A second-order differential equation is crucial to understanding the motion of the chain. In this problem, it is derived considering that the chain moves under the influence of gravity with no friction.
The differential equation is given as:
d^2y/dt^2 = (2g/l) * ydy/dt = rate of change of yThis equation is key because it describes how the difference in length, y, changes over time (t). It gives a mathematical representation of the chain's motion.Higher-order differential equations often arise in physics problems involving motion, as they account for acceleration (second derivative of position). Solving this equation involves finding a function y(t) that satisfies it.

hyperbolic functions

Hyperbolic functions appear naturally in the solution of second-order differential equations like the one described above. These functions, analogous to trigonometric functions, help simplify the problem.
When solving the differential equation:
ed^2y/dt^2 = (2g/l) * y,the general solution involves hyperbolic sine and cosine functions:y = A * cosh(t * sqrt(2g/l)) + B * sinh(t * sqrt(2g/l)).
The constants A and B depend on the initial conditions. Applying the given initial conditions y = y_0 when t = 0, we find that B = 0, simplifying our function to:
y = y_0 * cosh(t * sqrt(2g/l)).
In physics, hyperbolic functions frequently model behaviors in systems with linear restoring forces or are influenced by exponential growth or decay.