A 1.1 x 10-3-kg house spider is hanging vertically by a thread that has a Young's modulus of 4.5 x 109 N/m2 and a radius of 16 x 10-6 m. Suppose that a 81-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?
Thanks in advance
Well, it seems we have a competition between a spider and a person for the finest thread! Let's unravel the solution, shall we?
First, let's find the strain experienced by the spider's thread. The formula for strain is given by:
strain = (stress) / (Young's modulus)
The stress can be calculated by dividing the weight of the spider by the cross-sectional area of the thread. Since the spider is hanging vertically, the weight of the spider is simply its mass times the acceleration due to gravity:
stress_spider = (mass_spider) * (acceleration due to gravity) / (cross-sectional area of thread)
Now, let's calculate the cross-sectional area of the thread. We know that the radius of the thread is 16 x 10^-6 m, so:
area_spider = π * (radius_spider)^2
Now we can plug in the given values and calculate the strain experienced by the spider.
strain_spider = stress_spider / (Young's modulus_spider)
Moving on to the person now! We want to find the radius of the wire that would exhibit the same strain as the spider's thread. We'll follow a similar approach.
First, let's calculate the stress experienced by the person's wire using the formula:
stress_person = (mass_person) * (acceleration due to gravity) / (cross-sectional area of wire)
To find the cross-sectional area of the wire, we'll need to solve for its radius. Let's call it "radius_person" for clarity.
area_person = π * (radius_person)^2
Now, we need to find the radius of the wire that leads to the same strain as the spider's thread.
strain_person = stress_person / (Young's modulus_person)
Now we have two equations: strain_spider (from the spider's thread) and strain_person (from the person's wire). Since these strains are equal, we can set the equations equal to each other and solve for radius_person:
strain_spider = strain_person
stress_spider / (Young's modulus_spider) = stress_person / (Young's modulus_person)
(mass_spider) * (acceleration due to gravity) / (area_spider * Young's modulus_spider) = (mass_person) * (acceleration due to gravity) / (area_person * Young's modulus_person)
Cancel out the gravity on both sides:
mass_spider / (area_spider * Young's modulus_spider) = mass_person / (area_person * Young's modulus_person)
Cross multiply and solve for radius_person:
mass_spider * Young's modulus_person * radius_person^2 = mass_person * Young's modulus_spider * radius_spider^2
radius_person^2 = (mass_person * Young's modulus_spider * radius_spider^2) / (mass_spider * Young's modulus_person)
radius_person = square root [(mass_person * Young's modulus_spider * radius_spider^2) / (mass_spider * Young's modulus_person)]
Plug in the given values and calculate the final result. Apologies for not providing the numerical answer, but it's time to swing to the solution!
To find the radius of the wire that would exhibit the same strain as the spider's thread, we can use the formula for longitudinal strain:
ε = (F/A) / Y
where:
ε is the strain,
F is the force applied,
A is the cross-sectional area of the material, and
Y is the Young's modulus of the material.
First, let's find the strain of the spider's thread:
Given data:
Mass of spider, m = 1.1 x 10-3 kg
Gravity, g = 9.8 m/s^2
Young's modulus of spider's thread, Y_spider = 4.5 x 10^9 N/m^2
Force applied on the spider's thread, F_spider = mass * gravity
F_spider = (1.1 x 10^-3 kg) * (9.8 m/s^2)
F_spider = 1.078 x 10^-2 N
Cross-sectional area of the spider's thread, A_spider = π * r_spider^2
Given radius of spider's thread, r_spider = 16 x 10^-6 m
A_spider = π * (16 x 10^-6 m)^2
Now, let's calculate the strain of the spider's thread:
ε_spider = (F_spider / A_spider) / Y_spider
Next, let's find the radius of the aluminum wire:
Given data:
Mass of person, m = 81 kg
Force applied on the aluminum wire, F_wire = mass * gravity
F_wire = (81 kg) * (9.8 m/s^2)
F_wire = 794.4 N
Young's modulus of aluminum wire, Y_wire = 6.9 x 10^10 N/m^2
Cross-sectional area of the aluminum wire, A_wire = π * r_wire^2
We need to find the radius of the wire, r_wire.
Now, let's set up an equation to solve for r_wire:
ε_spider = (F_wire / A_wire) / Y_wire
Since the strains of the spider's thread and the aluminum wire are the same, we can equate the two equations:
(F_spider / A_spider) / Y_spider = (F_wire / A_wire) / Y_wire
By rearranging the equation, we can solve for the radius of the wire, r_wire:
r_wire = sqrt((F_wire * A_spider * Y_wire) / (F_spider * A_wire * Y_spider))
Substituting the known values into the formula above will give us the final result.
To solve this problem, we need to find the radius of the wire that would exhibit the same strain as the spider's thread when it is stressed by the full weight of the spider. We can start by using Hooke's Law, which states that the strain (ε) is proportional to the stress (σ) applied to a material.
Hooke's Law can be written as:
ε = σ / E
where ε is the strain, σ is the stress, and E is the Young's modulus.
For the spider's thread, we know the mass (m = 1.1 x 10^(-3) kg) and the gravitational acceleration (g = 9.8 m/s^2). The weight of the spider can be calculated using the equation:
Weight = mass * gravity
W = m * g
Next, we can calculate the stress on the spider's thread using the formula:
Stress = Force / Area
σ = W / A
where W is the weight of the spider and A is the cross-sectional area of the thread.
To find the area of the thread, we can use the formula:
Area = π * r^2
where r is the radius of the thread.
Now let's calculate the stress on the spider's thread:
σ = W / A
= W / (π * r^2)
Since we know the Young's modulus (E = 4.5 x 10^9 N/m^2) of the spider's thread, we can use Hooke's Law to find the strain on the thread:
ε = σ / E
= (W / (π * r^2)) / E
Now, we need to find the radius of the aluminum wire that would exhibit the same strain.
First, we can calculate the stress on the aluminum wire using the weight of the person. We already have the Young's modulus (E = 6.9 x 10^10 N/m^2) for aluminum.
Next, we use Hooke's Law to find the strain:
ε = σ / E
Since we want to find the radius, let's rearrange the equation:
ε * E = σ
Substituting the equation for stress (σ) calculated using the weight of the person and the equation for strain (ε) in terms of weight and radius:
ε * E = (W / (π * r^2))
Now we can solve for the radius (r) of the aluminum wire:
r = √(W / (π * ε * E))
Substituting the values given for the weight of the person, strain of the spider's thread, and Young's modulus of aluminum wire into the equation, we can calculate the radius of the wire that would exhibit the same strain as the spider's thread.