Problem 61 - Physics

Question

A \(1.0 \times 10^{-3}-\mathrm{kg}\) house 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?

Step-by-Step Solution

Answer

The radius of the wire is determined by equalizing strain conditions using the given Young's modulus and forces applied by the spider and the person. This requires solving for the radius with the computed strain and stress relationship.

Step 1: Understand the Given Values

We have a spider with mass \(1.0 \times 10^{-3}\, \text{kg}\) hanging from a thread with Young's modulus \(E_s = 4.5 \times 10^{9}\, \text{N/m}^2\) and radius \(r_s = 13 \times 10^{-6}\, \text{m}\). We also have a person with mass \(m_p = 95\, \text{kg}\) hanging from an aluminum wire. We need to find the radius of this wire, \(r_w\), such that both the spider's thread and the wire experience the same strain.

Step 2: Calculate the Force Exerted by the Spider

The force exerted by the spider is equivalent to its weight, which can be calculated using the formula \(F = mg\). Where \(m = 1.0 \times 10^{-3}\, \text{kg}\) and \(g = 9.81\, \text{m/s}^2\) is the acceleration due to gravity.\[F_s = (1.0 \times 10^{-3}) \times 9.81 = 9.81 \times 10^{-3}\, \text{N}\]

Step 3: Calculate the Stress on the Spider's Thread

Stress is defined as force per unit area. The area, \(A\), is related to the radius, \(r_s\), by the formula \(A = \pi r_s^2\).\[A_s = \pi (13 \times 10^{-6})^2\]Then the stress, \(\sigma_s\), on the spider's thread is calculated as:\[\sigma_s = \frac{F_s}{A_s} = \frac{9.81 \times 10^{-3}}{\pi (13 \times 10^{-6})^2}\]

Step 4: Relate Stress to Strain

Young's modulus, \(E\), relates stress to strain: \(E = \frac{\sigma}{\varepsilon}\). Therefore, the strain \(\varepsilon_s\) can be calculated by rearranging the equation:\[\varepsilon_s = \frac{\sigma_s}{E_s}\]

Step 5: Apply the Same Strain to the Aluminum Wire

For the aluminum wire, the strain \(\varepsilon_w\) is equal to \(\varepsilon_s\) since both the wire and the spider's thread experience the same strain. Therefore, \(\varepsilon_w = \varepsilon_s\). The Young's modulus of aluminum \(E_w = 70 \times 10^{9}\, \text{N/m}^2\) (reference value). To find the stress \(\sigma_w\) on the aluminum wire, use:\[\sigma_w = E_w \cdot \varepsilon_s\]

Step 6: Calculate the Force Exerted by the Person

The force exerted by the person hanging on the aluminum wire is \(F_p = m_p \cdot g\).\[F_p = 95 \times 9.81 = 931.95\, \text{N}\]

Step 7: Determine the Radius of the Aluminum Wire

Using the stress formula \(\sigma_w = \frac{F_p}{A_w}\), where \(A_w = \pi r_w^2\), we solve for \(r_w\) by rearranging:\[\sigma_w = \frac{931.95}{\pi r_w^2}\]Equating \(\sigma_w\) from Step 5 gives:\[E_w \cdot \varepsilon_s = \frac{931.95}{\pi r_w^2}\]Solving this equation, we can determine the radius \(r_w\) for the aluminum wire.

Key Concepts

Stress and Strain

In the world of physics, stress and strain are key concepts that describe how materials deform under different forces. **Stress** is essentially the internal force that a material experiences per unit area when subjected to an external force. It is calculated using the formula \( \text{stress} = \frac{F}{A} \), where \( F \) is the force applied, and \( A \) is the cross-sectional area.
On the other hand, **strain** is a measure of how much a material deforms in response to the stress it experiences. It is defined as the change in length divided by the original length, noted as \( \text{strain} = \frac{\Delta L}{L_0} \). Stress, measured in Pascals (N/m²), is necessary for calculating strain because the two are linked through Young's modulus. Understanding stress and strain helps us predict how materials will behave under different loading conditions.

Mechanical Properties

Mechanical properties are attributes that describe the behavior of materials under various forces. These properties determine how a material resists deformation and failure when forces are applied. Key mechanical properties include:

• **Strength**: The ability to withstand an applied force without failure.
• **Elasticity**: The capacity to return to its original shape after the removal of a load.
• **Plasticity**: The ability to undergo permanent deformation after the yield point is surpassed.
• **Hardness**: The resistance to scratching, indentation, or penetration.
• **Toughness**: The ability to absorb energy before fracturing.

These properties are crucial in selecting materials for various applications. Engineers and designers analyze them to ensure safety and functionality in structural applications like building bridges or skyscrapers. In the problem above, Young's modulus, a measurement of a material's elasticity, is central to determining how the spider's thread and the aluminum wire behave under stress.

Elasticity of Materials

Elasticity is the quality of a material to return to its original shape and size after being stretched or compressed. Young's modulus, denoted by \( E \), is a fundamental mechanical property that quantifies elasticity. It relates stress and strain through the equation \( E = \frac{\text{stress}}{\text{strain}} \).
When a material is subject to a stress within its elastic limit, it stretches proportionally. Once the stress is removed, the material returns to its original shape. The higher the modulus, the stiffer the material, indicating greater resistance to deformation.
In the exercise, the comparison between the spider's thread and the aluminum wire hinges on their elastic properties. Despite vastly different loads—spider versus person—the materials must display equal strain, facilitated by their respective Young's moduli. Understanding elasticity helps describe why some materials are more suitable than others for specific stress conditions. It also guides material choice in industries ranging from construction to manufacturing.