Young's Modulus: maximum depth of mine

[mmylo]

Homework Statement

Although human beings have been able to fly hundreds of thousands of miles into outer space, getting inside the Earth has proven much more difficult. The deepest mines ever drilled are only about 10 miles deep. To illustrate the difficulties associated with such drilling, consider the following: The density of steel is about 7900 kilograms per cubic meter, and its breaking stress, defined as the maximum stress the material can bear without deteriorating, is about 2.0x10^9 pascals. What is the maximum length of a steel cable that can be lowered into a mine? Assume that the magnitude of the acceleration due to gravity remains constant at 9.8 meters per second per second.

Homework Equations

Stress=F / A
Y=( F / A ) / ( delta L / L )

The Attempt at a Solution

I really don't know where to begin.

 

[PhanthomJay]

Homework Statement

Although human beings have been able to fly hundreds of thousands of miles into outer space, getting inside the Earth has proven much more difficult. The deepest mines ever drilled are only about 10 miles deep. To illustrate the difficulties associated with such drilling, consider the following: The density of steel is about 7900 kilograms per cubic meter, and its breaking stress, defined as the maximum stress the material can bear without deteriorating, is about 2.0x10^9 pascals. What is the maximum length of a steel cable that can be lowered into a mine? Assume that the magnitude of the acceleration due to gravity remains constant at 9.8 meters per second per second.

Homework Equations

Stress=F / A
Y=( F / A ) / ( delta L / L )

The Attempt at a Solution

I really don't know where to begin.

mmylo, welcome to PF!
You don't need to know Young's modulus or deformations to solve this problem. It is asking you to find the maximum length of the cable such that the maximum stress, due to the cables weight, does not exceed the given value of the maximum stress allowed. The cable's weight is a function of its weight density, length, and area. It's stress is just F/A. Does this give you a clue to solve the problem?

 

[thomate1]

Young's Modulus is a measure of a material's stiffness or elasticity and is often used to determine the amount of stress a material can withstand before breaking. In this case, we can use Young's Modulus to calculate the maximum length of a steel cable that can be lowered into a mine.

To begin, we need to use the formula Y=( F / A ) / ( delta L / L ) , where Y is Young's Modulus, F is the force being applied, A is the cross-sectional area of the cable, delta L is the change in length, and L is the original length of the cable.

In this scenario, the force being applied is the weight of the cable, which can be calculated using the density of steel and the maximum depth of the mine. The cross-sectional area of the cable can be calculated using the diameter of the cable.

Once we have these values, we can plug them into the formula and solve for the maximum length of the cable. This will give us an estimate of the maximum depth that the cable can be lowered into the mine without breaking.

However, it's important to note that this calculation does not take into account other factors such as the strength of the cable itself and the conditions inside the mine. Therefore, it is important for engineers and scientists to consider all of these factors when determining the maximum depth of a mine.