A piano wire has length L and mass M. If its fundamental frequency is f, its tension is:
i got T=(lambda*f)^2m/L, but their answer is 4LMF^2 and i cant figure out how to get that
the wavelength of the fundamental is twice the length of the wire.
L = n lambda/2 where n=1 for fundamental
L = lambda/2
but lambda = vT = v/f where v is speed
so
L = v/2f
now v = sqrt (TL/M)
so
2 f L = sqrt T L/M
4 f^2 L^2 = T L/M
T = 4 L M f^2
To determine the tension in a piano wire using the fundamental frequency, you can start with the equation for the speed of a wave:
v = λf
Where:
- v is the speed of the wave
- λ (lambda) is the wavelength of the wave
- f is the frequency of the wave
For a piano wire, the speed of the wave is related to its tension (T), mass per unit length (M/L), and length (L) by the equation:
v = sqrt(T / (M/L))
To find the frequency of the wave, we can use the equation:
f = v / λ
Substituting the expression for v:
f = sqrt(T / (M/L)) / λ
Now we need to find an expression for the wavelength. In the case of the fundamental frequency, the wavelength is twice the length of the wire:
λ = 2L
Substituting this expression for λ:
f = sqrt(T / (M/L)) / (2L)
Rearranging the equation gives:
2Lf = sqrt(T / (M/L))
Squaring both sides of the equation:
(2Lf)^2 = T / (M/L)
Expanding the equation:
4L^2f^2 = T * (M/L)
Note that (M/L) is the linear mass density of the piano wire, which we can denote as μ:
4L^2f^2 = T * μ
Therefore, the correct expression for the tension in the piano wire is:
T = 4L^2f^2 * μ
Where:
- T is the tension in the piano wire
- L is the length of the piano wire
- f is the fundamental frequency of the piano wire
- μ is the linear mass density of the piano wire (mass per unit length, M/L)
The correct expression for the tension T in the piano wire is actually T = (lambda * f)^2 * m / L, where lambda is the wavelength of the fundamental frequency and m is the mass of the wire.
To derive the given answer of T = 4LMf^2, let's go step-by-step:
1. Start with the equation for the speed of a wave on a string:
v = sqrt(T / mu)
where v is the velocity of the wave, T is the tension in the string, and mu is the linear mass density (mass per unit length) of the string.
2. The velocity of the wave can also be expressed as the product of the wavelength (lambda) and the frequency (f):
v = lambda * f
3. Rearrange equation (2) to solve for lambda:
lambda = v / f
4. Substitute equation (3) into equation (1):
v = sqrt(T / mu)
lambda * f = sqrt(T / mu)
5. Square both sides of equation (4) to eliminate the square root:
(lambda * f)^2 = T / mu
6. Multiply both sides of equation (5) by mu to isolate T:
T = (lambda * f)^2 * mu
7. Substitute the given linear mass density, mu = m / L, into equation (6):
T = (lambda * f)^2 * (m / L)
Hence, the correct expression for the tension T in the piano wire is T = (lambda * f)^2 * m / L, as you initially stated. The answer they provided, 4LMf^2, seems to be incorrect.