A 1.0 x 10-3-kg house spider is hanging vertically by a thread that has a Young's modulus of 4.2 x 109 N/m2 and a radius of 18 x 10-6 m. Suppose that a 61-kg person is hanging vertically on an aluminum (Young's modulus 6.9 x 1010 N/m2) 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?
this aint right sunny
To find the radius of the wire that would exhibit the same strain as the spider's thread, we can use Hooke's Law and the formula for strain.
1. Start by calculating the stress on the spider's thread using the formula:
stress = force / area
The force is the weight of the spider, which is given by:
force = mass x gravitational acceleration = 1.0 x 10^(-3) kg x 9.8 m/s^2
The area of the thread can be approximated as the cross-sectional area of a cylinder, so:
area = π x radius^2
Substitute the values into the formula:
stress_spider = (1.0 x 10^(-3) kg x 9.8 m/s^2) / (π x (18 x 10^(-6))^2)
2. Calculate the strain on the spider's thread using the formula:
strain = stress / Young's modulus
Substitute the values into the formula:
strain_spider = stress_spider / (4.2 x 10^9 N/m^2)
3. Now, find the radius of the wire that would exhibit the same strain as the spider's thread. This can be done by rearranging the formula for stress and substituting strain_spider and Young's modulus for the wire:
stress_wire = strain_spider x (6.9 x 10^10 N/m^2)
Substitute the values into the formula:
stress_wire = strain_spider x (6.9 x 10^10 N/m^2) = (61 kg x 9.8 m/s^2) / (π x radius_wire^2)
Rearrange the formula to solve for radius_wire:
radius_wire = sqrt((61 kg x 9.8 m/s^2) / (π x strain_spider x (6.9 x 10^10 N/m^2)))
4. Calculate the radius of the wire using the given values:
radius_wire = sqrt((61 kg x 9.8 m/s^2) / (π x strain_spider x (6.9 x 10^10 N/m^2)))
Substitute the calculated value of strain_spider from step 2 into the formula to find the radius of the wire.
To find the radius of the wire that would exhibit the same strain as the spider's thread when stressed by the full weight of the spider, we can use Hooke's Law, which states that the strain is directly proportional to the stress applied to a material.
The formula for strain is given by:
strain = (change in length) / (original length)
The stress in a material can be calculated using the formula:
stress = force / area
For the spider's thread, the stress is equal to the weight of the spider, as it is hanging vertically. Thus, we can calculate the stress in the spider's thread using the formula:
stress_spider = (weight of spider) / (cross-sectional area of thread)
Similarly, for the person hanging on the aluminum wire, the stress is equal to the weight of the person. Therefore, we can calculate the stress in the aluminum wire using the formula:
stress_aluminum = (weight of person) / (cross-sectional area of wire)
Since we want to find the radius of the wire that exhibits the same strain as the spider's thread, we need to equate the two stresses and solve for the unknown radius of the aluminum wire.
Here's the step-by-step solution:
1. Calculate the cross-sectional area of the spider's thread:
- For a thread with a circular cross-section, the area is given by the formula:
area_thread = π * (radius_thread)^2
2. Calculate the stress in the spider's thread:
- Use the formula:
stress_spider = (weight of spider) / (area_thread)
3. Calculate the strain in the spider's thread:
- Since the thread is hanging vertically, the strain is equal to the stress:
strain_spider = stress_spider
4. Calculate the cross-sectional area of the aluminum wire using the stress and Young's modulus:
- Use the formula:
stress_aluminum = (weight of person) / (area_aluminum)
area_aluminum = (weight of person) / (stress_aluminum)
5. Calculate the radius of the aluminum wire:
- Use the formula for the cross-sectional area of a wire:
area_aluminum = π * (radius_aluminum)^2
Solve for the unknown radius_aluminum:
radius_aluminum = √(area_aluminum / π)
6. Substitute the calculated stress for the aluminum wire in step 4 into the expression for strain in the spider's thread from step 3.
- Set the expressions for strain_spider and strain_aluminum equal to each other and solve for the unknown radius_aluminum.
7. Determine the radius of the aluminum wire that exhibits the same strain as the spider's thread when stressed by the full weight of the spider.
Remember to convert the given values to the appropriate units and perform any necessary calculations to obtain a precise numerical result.
σ =Eε
σ=Weight/A
σ₁ =mg/πr²= E₁ε
σ₂=Mg/πR² =E₂ε
(mg/πr²):(Mg/πR²) = (E₁ε):( E₂ε)
R =sqrt[ r²•M•E₁/m•E₂]=….