Final answer:
To calculate the extension produced, we can use Hooke's law and the formula for the spring constant of a wire. The minimum diameter of the wire can be found by comparing the maximum stress it can withstand to the stress produced by the hanging mass. We get, r = sqrt(5 / (Stress * π)).
Step-by-step explanation:
To calculate the extension produced by the hanging mass, we can use Hooke's law. Hooke's law states that the extension of a spring or wire is directly proportional to the applied force:
F = k * Δx
Where F is the force applied, k is the spring constant (a measure of how stiff the wire is), and Δx is the extension produced.
First, we need to calculate the spring constant k. The formula for the spring constant of a wire is:
k = (π * r^2 * E) / L
Where r is the radius of the wire, E is the Young's modulus of the material, and L is the length of the wire.
Substituting the given values into the formula, we have:
k = (π * (0.0025)^2 * E) / 2
Next, we can use Hooke's law to find the extension produced:
Δx = F / k
Substituting the given values, we have:
Δx = 5 / ((π * (0.0025)^2 * E) / 2)
Now, to find the minimum diameter of the wire so that its elastic limit is not exceeded, we need to compare the maximum stress the wire can withstand to the stress produced by the hanging mass. The stress is given by the formula:
Stress = F / (π * r^2)
Substituting the given values, we have:
Stress = 5 / (π * (0.0025)^2)
To find the minimum diameter, we can rearrange the stress formula as follows:
r^2 = F / (Stress * π)
r = sqrt(F / (Stress * π))
Substituting the given values, we have:
r = sqrt(5 / (Stress * π))