A piano wire with mass 3.00 g and length 80.0 cm is stretched with a tension of 25.0 N. A wav

A piano wire with mass 3.00 g and length 80.0 cm is...

A piano wire with mass $3.00 \mathrm{~g}$ and length $80.0 \mathrm{~cm}$ is stretched with a tension of $25.0 \mathrm{~N}$. A wave with frequency $120.0 \mathrm{~Hz}$ and amplitude $1.6 \mathrm{~mm}$ travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power if the wave amplitude is halved?

Key Concepts

Linear Mass Density

Linear mass density is the mass per unit length of a string or wire and is a crucial parameter in characterizing wave motion on a string. It affects how fast waves move along the string because the wave speed on a stretched string is inversely proportional to the square root of the linear mass density.

Wave Speed on a Stretched String

The wave speed on a stretched string is determined by the tension in the string and its linear mass density, following the relation v = ?(T/?). This concept is key to understanding how mechanical disturbances propagate along the string.

Average Power of a Wave

The average power transmitted by a wave on a string is related to the energy carried by the wave and the rate at which this energy is transported along the medium. For a sinusoidal wave, the power depends on factors such as the linear mass density, wave speed, angular frequency, and amplitude.

Dependence of Wave Power on Amplitude

In harmonic waves, the energy (and thus the power) is proportional to the square of the amplitude. This means that if the amplitude of the wave is reduced (for example, halved), the average power transmitted decreases by a factor equal to the square of the change (in this case, a reduction by a factor of four).

Transcript

00:01 So this question is asking for the average power, and we can say that the average power, we can write an equation for the average power.

00:08 This is going to be 1 over 2, and then it will be mu v.

00:19 Omega squared a square, a squared.

00:21 This is simply the equation for the average power of a wave, rather of a string, and then a wave traveling on a string.

00:32 And then the omega of course equals 2 pi f.

00:39 We know that to mu is going to be equal to m over l.

00:43 And then we should find the linear density first.

00:46 So m over l this is going to be equal to this is a 3 gram string or a 0 .003 kilogram string divided by 0 .8 meters or 80 centimeters.

01:01 And this is going to be 0 .00375 kilograms per meter.

01:10 So we always want to use si units always, always, always.

01:14 You want to do kilograms, meters, seconds, always, si units.

01:20 And then we know the amplitude is going to be 0 .00160 meters.

01:27 And then we can find the angular frequency 2 pi f this is going to be equal to 2 pi times 120 hertz again this is going to be 753 .98 radians per second at this point we know that rather we need to find the velocity so velocity will equal the force the force tension divided by the linear density.

02:04 We have the force tension so we can simply write it 25 newtons divided by 0 .00375.

02:15 This is going to be equal to 801 .65 meters per second.

02:22 And at this point we have all of our variables to tackle the average power.

02:28 So the average power it's going to be equal to 1 over 2 times mu, so 0 .00375 times the velocity 8 .1 .65...

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