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Chapter 10: Problem 69

\(\mathrm{A} 1.0 \times 10^{-3}-\mathrm{kg}\) spider is hanging vertically by a thread that has a Young's modulus of \(4.5 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}\) and a radius of \(13 \times 10^{-6} \mathrm{m}\) . Suppose that a \(95-\mathrm{kg}\) person is hanging vertically on an aluminum wire. What is the radius of the wire that would exhibit the same strain as the spider's thread, when the thread is stressed by the full weight of the spider?

Short Answer

Expert verified
Calculate the strain experienced by the spider's thread and recalculate the stress on an aluminum wire to find its radius, given equal strain conditions.

Step by step solution

01

Understanding the Problem

The problem requires us to find the radius of an aluminum wire that experiences the same strain as a spider's thread when each is subjected to their respective weights. We have information about Young's modulus for the spider's thread and need similar properties for aluminum to solve this.
02

Calculate the Stress on Spider's Thread

Stress is defined as force per unit area. The force exerted by the spider is its weight, given by:\[ F = m imes g = 1.0 \times 10^{-3} \times 9.8 \, \mathrm{N} \]The area of the spider's thread is:\[ A = \pi r^2 = \pi \times (13 \times 10^{-6})^2 \, \mathrm{m}^2 \]Thus, the stress \( \sigma \) on the thread is:\[ \sigma = \frac{F}{A} \]
03

Calculate Strain in Spider's Thread

Strain \( \varepsilon \) is given as stress divided by the Young's modulus \( Y \):\[ \varepsilon = \frac{\sigma}{Y} = \frac{F}{A \cdot Y} = \frac{1.0 \times 10^{-3} \times 9.8}{\pi \times (13 \times 10^{-6})^2 \times 4.5 \times 10^9} \]
04

Properties of Aluminum Wire

We need Young's modulus for aluminum, which is approximately \( Y_{\text{Al}} = 7.0 \times 10^{10} \mathrm{N/m^2} \).
05

Calculate Stress on Aluminum Wire

The force that the aluminum wire must bear is the weight of the person:\[ F = 95 \times 9.8 \, \mathrm{N} \]
06

Calculate Strain in Aluminum Wire

Strain must be the same as that experienced by the spider's thread. Thus, using the relation for strain previously calculated and solving for the new stress:\[ \varepsilon = \frac{\sigma_{\text{Al}}}{Y_{\text{Al}}} \] and\[ \sigma_{\text{Al}} = \varepsilon \times Y_{\text{Al}} \]
07

Finding Radius of Aluminum Wire

Plugging in the expression for stress on the aluminum wire, we equate stress and solve for the new area:\[ \sigma_{\text{Al}} = \frac{F}{A} = \frac{95 \times 9.8}{\pi r^2} = \varepsilon \times Y_{\text{Al}} \]Rearrange to solve for \( r \):\[ r = \sqrt{\frac{95 \times 9.8}{\pi \times \varepsilon \times Y_{\text{Al}}}} \]
08

Calculate Final Radius

Substitute the previously calculated \( \varepsilon \) to find the numerical value of \( r \) for the aluminum wire.

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Key Concepts

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

Stress Calculation
Stress is a measure of force applied over a unit area. It's like understanding how much pressure something is under. In this context, the stress on the spider's thread is calculated as the weight of the spider divided by the area of the thread.
The weight is found by multiplying the spider's mass by gravity:
  • Weight, \( F = m \times g \), where \( m = 1.0 \times 10^{-3} \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).
Once you have the force, you can calculate the area with the formula for the area of a circle:
  • Area, \( A = \pi r^2 \), using the radius given as \( 13 \times 10^{-6} \, \text{m} \).
Finally, stress \( \sigma \) is calculated by dividing the force by the area:
  • \( \sigma = \frac{F}{A} \).
This same concept applies to other materials, just with different forces and areas, showing a fundamental principle in engineering and physics.
Strain Calculation
Strain describes how much an object deforms under stress. It's a measure of the change in length divided by the original length. Here, we use the Young's modulus to relate stress and strain.
Strain \( \varepsilon \) is found using the formula:
  • \( \varepsilon = \frac{\sigma}{Y} \)
where \( Y \) is Young's modulus, representing the stiffness of a material.
In our example, the stress calculated from the spider's thread is divided by the given Young's modulus of \( 4.5 \times 10^{9} \, \text{N/m}^2 \). This provides the strain experienced by the thread. This same strain is what we're aiming to have in the aluminum wire.
Aluminum Properties
Aluminum is a versatile and widely-used metal known for its lightweight and strength properties. One key property is its Young's modulus, which is approximately \( 7.0 \times 10^{10} \, \text{N/m}^2 \) for aluminum.
This modulus tells us how aluminum will respond to stress. A higher Young's modulus means aluminum is stiffer compared to the spider's thread.
When a force is applied, understanding this modulus helps predict how much the aluminum wire will stretch. This property is essential for ensuring the aluminum wire's strain matches that of the spider's thread, providing insights into the wire's diameter calculations.
Thread and Wire Comparison
When comparing a spider's thread with an aluminum wire, we're essentially comparing materials with significantly different properties.
Both materials experience stress and strain, but due to their differing Young's moduli, they respond differently to the same pressure.
  • Threads like those of a spider are incredibly strong for their size, having a lower Young's modulus.
  • Aluminum wire, on the other hand, is used for heavier applications due to its higher stiffness and strength.
By aiming for the same strain in both, the problem helps us understand how altering dimensions (like radius) balances these different properties under similar conditions, which is a central consideration in material science and engineering.

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