Final answer:
To stretch the given copper wire 3.0 mm, a weight of approximately 3.467 Newtons, or a mass of about 0.353 kilograms, must be added to its free end.
Step-by-step explanation:
Calculating the Weight Needed to Stretch a Copper Wire:
To determine the weight needed to stretch the copper wire 3.0 mm, we need to apply the formula for the elastic deformation of materials, which relates the force applied, the cross-sectional area, Young's modulus, the initial length, and the elongation of the material:
F = (Y × A × ΔL) / L0
Where:
- F is the force (weight) needed,
- Y is Young's modulus for copper,
- A is the cross-sectional area of the wire,
- ΔL is the change in length (elongation), and
- L0 is the original length of the wire.
Young's modulus for copper (Y) is typically about 1.17 × 1011 N/m2. The cross-sectional area (A) can be calculated using the diameter (d):
A = π(d/2)2
Here, d = 1.0 mm = 1.0 × 10-3 meters. Therefore:
A = π(1.0 × 10-3m / 2)2 = 7.854 × 10-7 m2
With the elongation (ΔL) of 3.0 mm = 3.0 × 10-3 meters and the original length (L0) of 1.0 m, the force required to stretch the wire 3.0 mm can be calculated:
F = (1.17 × 1011 N/m2 × 7.854 × 10-7 m2 × 3.0 × 10-3 m) / 1.0 m = 3.467 Newtons
To find the weight in more familiar units, remember that weight (W) is given by W = mg, where g is the acceleration due to gravity (9.81 m/s2). Thus:
W ≈ 3.467 N * (1 kg · m/s2/N) * (1/9.81 m/s2) = 0.353 kg
This value represents the mass of the weight. To get the force (or weight in newtons), we simply use F = mg:
Weight needed = 0.353 kg * 9.81 m/s2 ≈ 3.467 Newtons