Standing waves on a string: 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

We can first find the wavelength of the standing wave using the formula:

wavelength = 2L/n

where L is the length of the wire and n is the harmonic number.

For the third harmonic (n=3), the wavelength is:

wavelength = 2(1.20 m)/3 = 0.80 m

Next, we can find the wave speed using the formula:

wave speed = frequency x wavelength

wave speed = 512 Hz x 0.80 m = 409.6 m/s

Finally, we can use the formula for the tension in a string under wave motion:

tension = (wave speed)^2 x mass per unit length

mass per unit length = density x cross-sectional area

cross-sectional area = πr^2

mass per unit length = (21.4 x 10^3 kg/m^3) x π(0.500 x 10^-6 m)^2 = 2.66 x 10^-7 kg/m

tension = (409.6 m/s)^2 x 2.66 x 10^-7 kg/m = 2.82 kN

Therefore, the answer is choice C.

Standing Waves on a String: a Platinum Wire That Is