A platinum wire that is 1.20 m long has a radius of 0.500 mm and is fixed at both 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? A) 4.00 kN B) 2.00 kN C) 2.82 kN D) 1.41 kN E) 1.00 kN

Question

Quizplus · Accepted Answer

The frequency of the 3rd harmonic is given by:

f₃ = 3v/2L

where v is the speed of sound in the wire and L is the length of the wire. We can solve for v:

v = 2Lf₃/3

The speed of sound in a wire is given by:

v = √(T/μ)

where T is the tension and μ is the linear density (mass per unit length) of the wire. We can solve for T:

T = μv²

Substituting v in the expression for v above, we get:

T = 4μL²f₃²/9

Substituting the values given:

μ = ρ(πr²) = 21.4 × 10⁻⁶(π(0.5 × 10⁻³)²) = 0.00848 kg/m
L = 1.20 m
f₃ = 512 Hz

T = 4(0.00848)(1.20)²(512)²/9 = 2.82 kN

Therefore, the tension in the wire is 2.82 kN, which is choice C.

A Platinum Wire That Is 1