Question

The steel used for piano wire has a breaking (tensile) strength pT of about 3×109N/m2 and a density rho of 7800kg/m3.

Part A

What is the speed c of a wave traveling down such a wire if the wire is stretched to its breaking point?

Express the speed of the wave numerically, in meters per second, to the nearest integer.

c =
m/s

Part B

Imagine that the wire described in the problem introduction is used for the highest C on a piano (C8≈4000Hz). If the wire is in tune when stretched to its breaking point, what must the vibrating length of the wire be?

Express the length numerically, in centimeters, using three significant figures.

L = cm

Answer 1

The speed of a wave in a steel piano wire at its breaking point is approximately 617 m/s. For a wire to be in tune with a frequency of 4000 Hz when stretched to its breaking point, the length of the vibrating wire must be about 7.71 cm.

Step-by-step explanation:

The speed of a wave on a string is given by the formula c = √(T/μ), where c is the speed, T is the tension, and μ is the linear mass density of the string. In this case, we are asked to calculate the wave speed at the breaking point of steel piano wire with a density of 7800 kg/m³ and tensile strength of 3×10⁹ N/m².

To find the wave speed at the breaking point, we use the formula c = √(pT/ρ), where pT is the tensile strength and ρ is the density. Plugging in the values, we get c = √(3×10⁹ N/m² / 7800 kg/m³). Calculating this gives a speed of c ≈ 617 m/s.

For Part B, we are to find the length L of wire that would resonate at a frequency of 4000 Hz when stretched to its breaking point. Using the formula L = c/(2f), where f is the frequency, we calculate L = 617 m/s / (2×4000 Hz), which gives L ≈ 0.0771 m or L ≈ 7.71 cm when converted to centimetres.

Answer 2

To determine the wave speed at the breaking point and the length of the wire for a 4000Hz frequency, knowledge of the wire's diameter is required to calculate its area and linear mass density, which are necessary to solve both parts of the question.

Step-by-step explanation:

To calculate the speed of a wave traveling down a piano wire at its breaking point, we use the equation for wave speed on a string, which is c = √(T/μ), where T is the tension (equal to the breaking strength pT) and μ is the linear mass density of the wire. The linear mass density of the wire can be found using μ = rho/A, where rho is the density of the material and A is the cross-sectional area of the wire.

Given that the breaking strength pT is approximately 3×109 N/m2 and the density rho is 7800 kg/m3, and assuming a cylindrical shape of the wire, we need to know the diameter of the wire to calculate the area A and subsequently the linear mass density μ.

Without the diameter of the wire, we cannot provide a numerical answer for the speed of the wave c or calculate the length L of the wire for a frequency of C8 which is approximately 4000Hz.