A wire of diameter 0.65mm and length 6.0m stretches 10mm when 5.5kg is hung from it. What is the Young's modulus for the wire and what is the effective spring constant for the stretching wire

To find the Young's modulus (E) and the effective spring constant (k) for the wire, we will use Hooke's Law and the formula for Young's modulus.

Hooke's Law states that the force (F) applied to a spring or wire is directly proportional to the extension or deformation (Δx) of the wire. Mathematically, this relationship can be expressed as:

F = k * Δx

where F is the force applied, k is the spring constant, and Δx is the change in length.

In this case, the wire stretches by 10mm (or 0.01m) when a mass of 5.5kg is hung from it. We need to find the spring constant (k) and then use it to calculate the Young's modulus (E).

Let's start by calculating the spring constant (k):

Step 1: Find the force applied (F)
The weight of the hanging mass can be calculated using the formula: weight = mass * gravitational acceleration. The gravitational acceleration on Earth is approximately 9.81 m/s^2.

F = 5.5kg * 9.81m/s^2
F = 53.955 N

Step 2: Calculate the spring constant (k)
Using Hooke's Law, we can rearrange the formula to solve for k:

k = F / Δx

k = 53.955 N / 0.01m
k = 5395.5 N/m

Now we have the spring constant (k). Next, we can calculate the Young's modulus (E) using the following formula:

E = (F / A) / (∆L / L)

where A is the cross-sectional area of the wire, ∆L is the change in length of the wire, and L is the original length of the wire.

Step 3: Calculate the cross-sectional area (A)
The formula for the area of a wire is:

A = π * (d/2)^2

where d is the diameter of the wire. The diameter is given as 0.65mm, which we can convert to meters:

d = 0.65mm = 0.00065m

Now we can calculate the cross-sectional area:

A = π * (0.00065m / 2)^2
A = π * (0.000325m)^2
A = 3.14159 * 1.05625e-7 m^2
A ≈ 3.32678e-7 m^2

Step 4: Calculate the Young's modulus (E)

E = (F / A) / (∆L / L)

E = (53.955 N / (3.32678e-7 m^2)) / (0.01m / 6.0m)
E = (53.955 N / 3.32678e-7 m^2) / 0.0016667
E ≈ 323014252 N/m^2 (or Pascal, Pa)

Therefore, the Young's modulus for the wire is approximately 323014252 Pa (or N/m^2).

The effective spring constant (k) for the stretching wire is already calculated in step 2, and it is approximately 5395.5 N/m.