Waves: find amplitude, given average power

[wayfarer]

Homework Statement

A sinusoidal transverse wave travels on a string. The string has length 7.60 m and mass 6.50 g. The wave speed is 34.0 m/s and the wavelength is 0.250 m.
If the wave is to have an average power of 55.0 W, what must be the amplitude of the wave?

Homework Equations

Average Power = 0.5 * sqrt( u F ) * w^2 * A^2 where:
u = density of string, F =force, w = angular velocity, A = amplitude

The Attempt at a Solution

I put in u = (6.50*10^-3)/(0.25) = 8.55 * 10^-4
F =v^2 * u = (34)^2 * (8.55 * 10^-4 )
w = 2*pi* f = 2*pi* v/lambda = 2*3.14*(34)/(0.25)
and solved for A, and got the wrong answer (A = 2.104 m).
Where have I gone wrong?

 

[alphysicist]

Hi wayfarer,

Homework Statement

A sinusoidal transverse wave travels on a string. The string has length 7.60 m and mass 6.50 g. The wave speed is 34.0 m/s and the wavelength is 0.250 m.
If the wave is to have an average power of 55.0 W, what must be the amplitude of the wave?

Homework Equations

Average Power = 0.5 * sqrt( u F ) * w^2 * A^2 where:
u = density of string, F =force, w = angular velocity, A = amplitude

The Attempt at a Solution

I put in u = (6.50*10^-3)/(0.25) = 8.55 * 10^-4

I think there is an error here; you are using the total mass of the string, so you need to put in the total length of the string (not the wavelength of the wave). But I guess that is just a typo in your post, because you have the right answer for $\mu$ so you did really divide by 7.6 m?

Can you give some more details (what you got for the intermediate values F, w, etc.). I did not get 2.104 m from your numbers, but it's difficult to tell where you might have gone wrong without more details.

 

[Moetasim]

The equation you used for average power is incorrect. The correct formula for average power in a transverse wave is P = (1/2) * u * v * A^2 * w^2, where u is the linear density of the string, v is the wave speed, A is the amplitude, and w is the angular frequency.

Using this equation, we can solve for the amplitude A by rearranging the equation to A = √(2P/(u * v * w^2)). Plugging in the given values, we get A = √(2*55.0/(8.55*10^-4 * 34.0 * (2*3.14*34.0/0.25)^2)) = 0.00795 m.

Therefore, the amplitude of the wave is 0.00795 m.