Question

A platinum wire that is 1.20 m long has a radius of 0.500 mm and is fixed at ends. In its third harmonic it vibrates at 512 Hz. The density of platinum is 21.4 × 103 kg/m3. What is the tension in the wire?

Answer 1

Answer:

2820 N

Explanation:

Let assume that

Ft = µ(2fL)^2

tension = µ(2*frequency*length)²  

wavelength = 2*length

In general, for a wire fixed at both ends,

λ_n = 2L/n

n is the harmonic number

v = f_n * λ_n = f_n * 2*L/n

= √(T/µ)

= √(T/(ρA))

so  

T = (f_n * 2 * L / n)² * ρA

Given that,

f_3 = 512 Hz,

L = 1.2 m,

n = 3,

ρ = 21.4 × 10⁻³ kg/m³,

and with the radius can easily find the area A

A = πr²

  = π(0.005)²

  = 7.85 ×10⁻⁵

When I plug these values in, I get tension T = 2820 N.