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

Answer:

73.8 N

Explanation:

The total volume is,

V = $\frac{m_Al}{P_Al} = \frac{m_copper}{P_copper}$

=  $\frac{10m}{(100)(2.6)} = \frac{90m}{(100)(8.9)}$

= 0.1396 m

The average  density is,

$p = \frac{m}{V}$

= $\frac{m}{0.1396}$

= 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