Question

If an alloyed guitar B string (147 Hz resonance) has an average diameter of .6 mm over an effective oscillating length of 65 cm, what is the tension in Newtons if the alloy is 10% aluminum and 90% copper? (assume the specific gravities of the metals are 2.6 and 8.9 grams per cubic centimeter, respectively)

Answer 1

73.8 N

Step-by-step explanation:

The total volume is,

V =
`(m_Al)/(P_Al) = (m_copper)/(P_copper)`

=
`(10m)/((100)(2.6)) = (90m)/((100)(8.9))`

= 0.1396 m

The average density is,

`p = (m)/(V)`

=
`(m)/(0.1396)`

= 7.169 g/cm³

The linear mass density is,

μ = pπr²

= (7.169 x 10⁹) (π (0.3 x 10⁻³)²)

= 2.026 x 10⁻³ Kg/m

The fundamental mode of length is,

L = λ/2

λ=2L

= 2 x 0.65

= 1.3 m

The speed of the wave is,

v = λf

= 1.3 m x 147 Hz

= 1.91 m/s

The tension is,

v = √T/ц

T = ц v²

= 2.026 x 10⁻³)(1.91 m/s)²

= 73.769N

73.8N