Question

Final naïve case: If the highest-pitch string on the piano is made of spring steel (density = 7800 kg/m3) with a diameter of 1/32" (= 0.794 mm), what will the linear density of such a string be (in kg/m)?

Answer 1

The linear density is
`K = 3.863 *10^(-3 ) \ kg/m`

Step-by-step explanation:

From the question we are told that

The density of steel is
`\rho = 7800 \ kg/m^3`

The diameter of the string is
`d = 0.794 \ mm = 7.94 *10^(-4) \ m`

The radius of the string is evaluated as
`r = (D)/(2) = (7.94 *10^(-4))/(2) = 3.97*10^(-4) \ m`

The volume of the string is mathematically evaluated as

`V = \pi * r ^2 * L`

Now assuming that the length of the string is L = 2 m

So

`V = 3.142 * (3.97 *10^(-4))^2 * (2)`

`V = 9.9041 *10^(-7) \ m^3`

Then the mass of the string would be

`m = \rho * V`

substituting value

`m = 7800*9.904 14 *10^(-7)`

`m = 7.73*10^(-3) \ kg`

Looking at the question we see that the unit of the linear density is
`(kg)/(m)`

Hence the linear density is evaluated as

`K = (m)/(L)`

substituting value

`K = (7.73 *10^(-3))/(2)`

`K = 3.863 *10^(-3 ) \ kg/m`