Final answer:
The mass that is causing the steel wire to stretch is approximately 1.37 kg.
Step-by-step explanation:
To find the mass that is causing the steel wire to stretch, we can use the formula for stress:
Stress = Force/Area
First, we need to find the force exerted by the weight:
Force = mass × acceleration due to gravity
Since the mass is unknown, we'll call it 'm'. The acceleration due to gravity is approximately 9.8 m/s². Therefore, the force is:
Force = m × 9.8 N
Now, let's calculate the cross-sectional area of the wire:
Area = π × (radius)²
The diameter of the wire is 2.1 mm, so the radius is half of that (1.05 mm = 0.00105 m).
Area = π × (0.00105 m)²
Finally, we can calculate the stress
Stress = Force/Area = (m × 9.8 N)/(π × (0.00105 m)²)
The given information is that the wire stretches by 0.058%, which means the strain is 0.00058. Stress is related to strain by Young's modulus (E):
Stress = E × Strain
Therefore, we can rearrange the equation to solve for the mass:
m = (Stress × Area)/(9.8 N)
Plugging in the values:
m = ((2.0 × 1011 N/m²) × (π × (0.00105 m)²))/(9.8 N)
Using a calculator, we find that the mass is approximately 1.37 kg.