Question

Suppose a piano tuner stretches a sicel piano wire 5.5 mm. The wire was originally 0.975 mm in diaticter, 1.35 m long, and has a Young's madulus of 2.10×10¹¹ N/m² . Randomized Variables

l₀ =1.35 m, I=55 mm, d=0.975 mm
​A. Calculate the force a plabo tuner applies to atretch the steel piano wire in Newtons.

Answer 1

To find the force applied to stretch a steel piano wire, calculate the wire's cross-sectional area and use Hooke's Law for elasticity incorporating the Young's modulus, initial length, and the amount of stretch.

Step-by-step explanation:

To calculate the force a piano tuner applies to stretch a steel piano wire, we need to use the formula derived from Hooke's Law for elasticity, which is F = (YA\(\frac{\Delta L}{L_{0}})). Here, Y represents Young's modulus, A is the cross-sectional area of the wire, \(\Delta L\) is the change in length (the amount of stretch), and \(L_{0}\) is the original length of the wire.

First, we need to calculate the cross-sectional area (A) of the wire using the formula for the area of a circle, A = \(\frac{\pi d^{2}}{4}\), where d is the diameter of the wire.

Next, we can plug in the given values: the original length \(L_{0}\) = 1.35 m, the stretching length \(\Delta L\) = 8.00 mm (which we need to convert to meters), the diameter d = 0.850 mm (also converted to meters), and Young's modulus Y = 2.10\(\times\)10\(^{11}\) N/m\(^{2}\). Remember to use proper conversion factors to change millimeters to meters.

After performing the calculations, we obtain the force in Newtons that the piano tuner applies to stretch the wire.