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ricmacas
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Homework Statement
One end of a horizontal string is attached to the wall, and the other end passes over a pulley. A sphere of an unknown material hangs on the end of the string. The string is vibrating with a frequency of 392 cycles per second. A container of water is raised under the sphere so that the sphere is completely submerged. In this configuration, the string vibrates with a frequency of 343 cycles per second.
What is the density of the sphere?
Homework Equations
[itex]v=\sqrt{T/\rho}[/itex]
[itex]v=\lambda * f[/itex]
[itex]W=\rho*V*g[/itex]
[itex]F=W-B=\rho*V*g - \rho_{H20}V_{i}g[/itex] (assuming W>B)
The Attempt at a Solution
Wave Perspective:
[itex]T_{1} = \rho * \lambda_{1}^{2} * f_{1}^{2}[/itex]
[itex]T_{2} = \rho * \lambda_{2}^{2} * f_{2}^{2}[/itex]
Mechanical Perspective:
[itex]T_{1} = W = \rho*Vg[/itex]
[itex]T_{2} = W - B = \rho*Vg - \rho_{H20}V_{i}g[/itex]
Thus:
[itex]T_{1} = W[/itex] then, assuming displaced volume = volume of the sphere,
[itex] V_{i}g = \lambda_{1}^{2} * f_{1}^{2} [/itex]
[itex]T_{2} = W - B = \rho*\lambda_{1}^{2} * f_{1}^{2} - \rho_{H20}*\lambda_{1}^{2} * f_{1}^{2}[/itex]
Now, I have arrived to the answer in the solutions (it's the right answer, but negative, I can't explain that either), but I had to assume that [itex]\lambda_{1}=\lambda_{2}[/itex]. Can someone explain me , if frequencies are different, why should I assume the wavelenght is the same before and after submerging the ball?
For those who want the answer:
4267 Kg m-3, which i obtained from [itex]T_{2}=T_{2} \Leftrightarrow \rho * \lambda^{2} * f_{2}^{2} = \rho\lambda^{2} * f_{1}^{2} - \rho_{H20}\lambda^{2} * f_{1}^{2}[/itex]
thus [itex]\rho = (-\rho_{H20}*f_{1}^{2})/ (f_{2}^{2} - f_{1}^{2})[/itex]
(The formula gives me a negative result though).
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