Mechanical Waves On a String - Speed, Amplitude, and Power

In summary, a string of mass 38.5g and length 5.60m under tension of 220N has a wave with frequency 178 Hz traveling on it. Using the equations for the speed of the wave and power, the correct values for the speed of the wave and amplitude of the wave are 179 m/s and 1.35 cm, respectively. However, there was an error in the calculation due to a conversion mistake, which resulted in incorrect values initially.
  • #1
Nickg140143
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Homework Statement


A string of mass 38.5g and length 5.60m is secured so that it is under tension of 220N. A wave with frequency 178 Hz travels on the string. Find the speed of the wave and the amplitude of the wave if it transmits power of 140 Watts.

The Given answers are: 179 m/s and 1.35 cm

Homework Equations


I believe these are the main equations that I can use to solve these problems

speed of wave
[itex]v=\sqrt{\frac{Tension}{\mu}},\mu = \frac{Mass}{Length}[/itex]

Power (from my notes)

[itex]P=\frac{\mu \times v \times \omega^2 \times A^2}{2},\omega = 2\pi f[/itex]

Power (from my book, think this is average power)

[itex]P=\frac{\sqrt{\mu \times F} \times \omega^2 \times A^2}{2},\omega = 2\pi f[/itex]

Are my notes correct? Are these equations for the same amount of power?

The Attempt at a Solution



Well, I'm given mass, length and tension of the string, and If my understanding is somewhat correct, the speed at which a wave moves through a medium is dependent only on the properties of the medium itself, and for this string, these properties are the mass, length, and tension

[itex] mass(M) = 38.5 g(grams) \longrightarrow .385 kg(kilograms)[/itex]
[itex] tension(T) = 220N(\frac{kgm}{s^2})[/itex]
[itex] length(L)=5.60m(meters)[/itex]

With this information, I can use my formula for velocity of the wave to find the speed of the wave.

I'll calculate [itex]\mu[/itex]
[itex]\mu = \frac{M}{L} \longrightarrow \mu = \frac{.385 kg}{5.60 m}=0.069\frac{kg}{m}[/itex]

Now [itex]v[/itex]
[itex]v=\sqrt{\frac{T}{\mu}}\longrightarrow v=\sqrt{\frac{220\frac{kgm}{s^2}}{0.069\frac{kg}{m}}}=\boxed{56.5\frac{m}{s}}[/itex]
Since I now have the velocity [itex]v[/itex], I should use the equation for power [itex]P[/itex], where [itex]P = 140 watts(\frac{kgm^2}{s^3})[/itex] and solve for A

[itex]P=\frac{\mu \times v \times \omega^2 \times A^2}{2},\omega = 2\pi f[/itex]

but before I can use this, I need [itex]\omega[/itex], which is [itex]\omega = 2\pi \times frequencey(f)[/itex]

[itex]\omega = 2\pi(rads) \times 178 \frac{1}{s}=1118.41\frac{1}{s}[/itex]now that I have [itex]\omega[/itex], I can solve for A

[itex]140\frac{kgm^2}{s^3}=\frac{(0.069\frac{kg}{m}) \times (56.5\frac{m}{s}) \times(1118.41\frac{1}{s})^2 \times A^2}{2}[/itex]

[itex]\frac{280\frac{kgm^2}{s^3}}{(0.069\frac{kg}{m}) \times (56.5\frac{m}{s}) \times(1118.41\frac{1}{s})^2}=A^2[/itex]

[itex]A^2=5.74\times 10^{-5}m[/itex] or [itex]0.0000574 m \longrightarrow \boxed{A=.0076m \rightarrow .76cm}[/itex]But as you can see, these answers do not correspond with those that are given. There are plenty of concepts that I'm struggling with here in physics, so I want to make sure I'm not straying too far off the path here.

any help would be greatly appreciated.
 
Last edited:
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  • #2
Found my problem...

[itex].385kg[/itex]

This mass should actually be

[itex].0385kg[/itex]

Looks like I simply messed up on a conversion.
 

FAQ: Mechanical Waves On a String - Speed, Amplitude, and Power

1. What factors affect the speed of mechanical waves on a string?

The speed of mechanical waves on a string is affected by two main factors: the tension of the string and the density of the medium through which the wave is traveling. Higher tension and lower density will result in a faster wave speed.

2. How does the amplitude of a mechanical wave on a string affect its energy?

The amplitude of a mechanical wave on a string is directly proportional to its energy. This means that the higher the amplitude, the more energy the wave carries. This can be seen in the increased height of a wave on a string as it carries more energy.

3. Can the speed of a mechanical wave on a string be changed?

Yes, the speed of a mechanical wave on a string can be changed by altering the tension of the string or the density of the medium. In addition, the type of wave (transverse or longitudinal) and the frequency of the wave can also affect its speed.

4. How does the power of a mechanical wave on a string relate to its amplitude?

The power of a mechanical wave on a string is directly proportional to the square of its amplitude. This means that doubling the amplitude of a wave will result in four times the power. This relationship can be seen in the increased loudness of a sound wave as its amplitude increases.

5. Can the amplitude of a mechanical wave on a string be changed without altering its speed?

Yes, the amplitude of a mechanical wave on a string can be changed without altering its speed by adjusting the energy input into the string. This can be done through changing the source of the wave, such as plucking a guitar string harder, or using a mechanical device to increase the amplitude without changing the speed of the wave.

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