Final answer:
To calculate the weight that must be added to the free end of the copper wire in order to stretch it by 3.0 mm, we can use Hooke's Law and Young's modulus. By rearranging the equations and substituting the given values, we can calculate the weight to be approximately 7.065 x 10^-13 N.
Step-by-step explanation:
To calculate the weight that must be added to the free end of the copper wire in order to stretch it by 3.0 mm, we need to consider the properties of the wire. The diameter of the wire is 1.0 mm, which means its radius is 0.5 mm or 0.0005 m. The length of the wire is 1.0 m.
The stretch of the wire can be found using Hooke's Law, which states that the extension of an elastic material is directly proportional to the force applied to it. The formula for Hooke's Law is:
F = k * x
Where F is the force (weight), k is the spring constant (which in our case is equivalent to the Young's modulus), and x is the stretch (extension).
The equation can be rearranged to solve for the weight:
F = k * x
W = k * x
Now we can calculate Young's modulus for copper, which is given by:
k = (F/A) / x
Where A is the cross-sectional area of the wire.
Given that the diameter of the wire is 1.0 mm, the radius is 0.5 mm or 0.0005 m. The cross-sectional area can be calculated using the formula:
A = π * r^2
Substituting the values, we get:
A = 3.14 * (0.0005)^2 = 7.85 x 10^-7 m^2
Now we can calculate Young's modulus:
k = (F / (7.85 x 10^-7 m^2)) / (3.0 x 10^-3 m) = F / 2.355 x 10^-10 N
We can rearrange this equation to solve for the weight:
W = k * x
W = 2.355 x 10^-10 N * 3.0 x 10^-3 m = 7.065 x 10^-13 N
Therefore, the weight that must be added to the free end of the copper wire in order to stretch it by 3.0 mm is approximately 7.065 x 10^-13 N.