Uniform Circular Motion frequency

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The lab focused on the relationship between the frequency of an object in uniform circular motion and factors such as force, radius, and mass. Observations indicated that force is proportional to the frequency squared, while both radius and mass are inversely proportional to the frequency squared. The derived equations for force, radius, and mass were presented, along with a need to combine these results into a single equation for frequency in terms of tension, radius, and mass. The goal is to align this combined equation with the established formula SIGMA FORCE = 4PIE^2mrf^2. Assistance is sought for combining the equations and addressing discrepancies in data.
DevilTemptations
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Yesterday I did a lab with the purpose of determining the relationship between the frequency of revolution of an object in uniform circular motion and
1)the magnitude of the force (causing the circular motion)
2)the radius of the circular path
3)the mass of the object.

Well by graphing my observations, I was able to determine that the force is proportional to the frequency squared. The radius is proportional to the inverse of the frequency squared. The mass is also proportional to the inverse of the frequency squared.

By determing the line of best fit of these three graphs, I got the equations

Force=0.2639(frequency) - 0.1176
Radius=2.3983(frequency) + 0.1209
Mass=0.0343(frequency) - 0.0011

I also converted my proportionality statements into equations
frequency= force / k square rooted
frequency = k / radius square rooted
frequency = k / mass square rooted

Apparently I have to combine my "three results" to obtain an equation for the frequency in terms of the tension(force), the radius and the mass. It should provide the same answers if I checked it with data points as the equation SIGMA FORCE = 4PIE^2mrf^2 (There is obviousy going to be some disrepancies but it should be a close result)

HOW DO I COMBINE EQUATIONS?? HELP! :cry:
 
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I am doing the same lab but my data is not correct what data did you get?
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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