Calculating Density of Immersed Weight using Vibrations and Fluid Statics

[boomboompoop]

Vibrations and Fluid Statics

One end of a horizonatal wire is fixed, while the other passes over a smooth pulley and has a heavy weight attached to it. The frequency of the fundamental note emitted when the wire is plucked is 392 Hz. When the weight is completely immersed in water, the new fundamental frequency is 343 Hz. Calculate the density of the weight.

 

[nrqed]

Vibrations and Fluid Statics

One end of a horizonatal wire is fixed, while the other passes over a smooth pulley and has a heavy weight attached to it. The frequency of the fundamental note emitted when the wire is plucked is 392 Hz. When the weight is completely immersed in water, the new fundamental frequency is 343 Hz. Calculate the density of the weight.

WHat have you done so far? Do you know how the weight of an object changes when it is immersed in a liquid?

 

[kyle8921]

I would approach this problem by first understanding the concepts of vibrations and fluid statics. Vibrations refer to the back and forth motion of an object, while fluid statics is the study of fluids at rest. In this scenario, the wire is undergoing vibrations when it is plucked, and the weight is being immersed in a fluid (water) at rest.

To calculate the density of the weight, we can use the equation for frequency of a vibrating string, which is given by f = (1/2L)√(T/μ), where L is the length of the string, T is the tension, and μ is the linear density of the string.

In this case, the length of the string remains constant, as well as the tension since the weight is attached to the end of the string. Therefore, the change in frequency is due to the change in the linear density of the string.

We can rearrange the equation to solve for μ, which gives us μ = (4f^2L^2)/T.

Substituting the given values, we get μ = (4(392 Hz)^2(L^2))/T. We can then use this value to calculate the density of the weight when it is immersed in water.

The new frequency, 343 Hz, can be plugged into the equation to solve for the new linear density, μ'. μ' = (4(343 Hz)^2(L^2))/T.

To find the density of the weight, we can use the fact that the density of water is 1000 kg/m^3. We can set the two expressions for μ and μ' equal to each other and solve for the density of the weight, ρ.

ρ = μ'/μ = (4(343 Hz)^2(L^2))/T / (4(392 Hz)^2(L^2))/T = (343 Hz)^2 / (392 Hz)^2 * 1000 kg/m^3

Therefore, the density of the weight is approximately 877 kg/m^3.

In conclusion, by understanding the principles of vibrations and fluid statics, we were able to use the given information to calculate the density of the weight when it is immersed in water. This type of problem showcases the interdisciplinary nature of science, where knowledge from different fields can be applied to solve a problem.