Buoy in water differential equation finding weight

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SUMMARY

The discussion centers on calculating the weight of a cylindrical buoy with an 18-inch diameter that vibrates with a period of 2.7 seconds when slightly depressed in water. The buoyancy force is determined using the formula: weight = density * volume, with water's density specified as 62.5 lb/ft³. The participant attempts to relate the period of vibration to the mass and spring constant using the equation T = 2π√(m/k) and explores the relationship between the spring constant and the distance stretched. The calculations lead to confusion regarding the correct height for volume, indicating a need for clarification on the application of these equations.

PREREQUISITES
  • Understanding of buoyancy principles and Archimedes' principle
  • Familiarity with the concept of simple harmonic motion
  • Knowledge of basic geometry for calculating volume of a cylinder
  • Proficiency in algebraic manipulation of equations
NEXT STEPS
  • Review the derivation of the buoyancy force formula in fluid mechanics
  • Study the relationship between period, mass, and spring constant in harmonic motion
  • Learn how to calculate the volume of a cylinder using the formula V = πr²h
  • Explore examples of similar problems involving oscillating objects in fluids
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Students in physics or engineering courses, particularly those studying fluid dynamics and harmonic motion, as well as educators seeking to clarify concepts related to buoyancy and oscillation.

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Homework Statement



A cylindrical buoy with diameter 18 in. floats in water with its axis vertical. When depressed slightly its period of vibration is found to be 2.7 seconds. Find the weight of the cylinder.

Homework Equations



I know that an object submerged in water is buoyed up by a force equal to the product of density and volume, and the density of water is given as 62.5 lb/ft3

The Attempt at a Solution



Treating it as a spring problem

Weight of buoyancy force = density * volume = (62.5 lb/ft3) * volume
Period = 2(pi) sqrt (m/k) = 2.7sec
k = mass / distance stretched
volume = (pi) (r)2h

Since Period (T) 2.7sec then is this valid?

2.7 = 2(pi) sqrt (m/k) --> (2.7/(2pi)2= (m/k)
and since k = m/d --> (2.7/(2pi)2 = d
so d = 0.1846578572 ?

So would this d become the height for volume? Because that gives me 46.98970118, which multiplied with 62.5 gives 2936.856324, which I know isn't right.
 
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