Final answer:
To calculate the force a piano tuner applies to stretch a steel piano wire, you need to use Hooke's Law and the formula for the spring constant. In this case, given the dimensions of the wire and the Young's modulus of steel, the force is approximately 26,400 N.
Step-by-step explanation:
To calculate the force a piano tuner applies to stretch a steel piano wire, we can use Hooke's Law.
Hooke's Law states that the force (F) applied to an object is directly proportional to the change in length (ΔL) of the object and the spring constant (k) of the object.
In this case, the piano wire is being stretched by 8.00 mm, so ΔL = 8.00 mm = 0.008 m.
The original length of the wire is given as 1.35 m. The spring constant can be calculated using the formula:
k = (π * (d/2)^2 * E) / L
where d is the diameter of the wire and E is the Young's modulus of the wire material.
The diameter of the wire is given as 0.850 mm = 0.00085 m. The Young's modulus of steel is approximately 2.0 x 10^11 N/m^2.
Plugging the values into the formula, we have:
k = (π * (0.00085/2)^2 * (2.0 x 10^11)) / 1.35
Calculating this gives us a spring constant (k) of approximately 3.30 x 10^6 N/m. Now we can use Hooke's Law to calculate the force (F) applied to stretch the wire:
F = k * ΔL
F = (3.30 x 10^6) * (0.008)
Calculating this gives us a force (F) of approximately 26,400 N.
Therefore, the correct answer is not provided in the options given.