Buoy in water differential equation finding weight

In summary, a cylindrical buoy with diameter 18 in. and a period of vibration of 2.7 seconds when slightly depressed has a weight of approximately 2936.856324 lbs based on the given density of water and the calculated volume. This calculation is based on treating the problem as a spring problem and using the equations for buoyancy force and period. However, there may be some uncertainty in the final result as the calculation for distance stretched is not entirely clear. Further clarification or confirmation may be needed for a more accurate calculation.
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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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Related to Buoy in water differential equation finding weight

1. What is a "Buoy in water differential equation"?

A "buoy in water differential equation" is a mathematical equation that describes the relationship between the buoyancy force acting on an object in water and the weight of the object. It takes into account factors such as the volume and density of the object, as well as the density of the water.

2. Why is it important to find the weight of a buoy in water?

Knowing the weight of a buoy in water is important for several reasons. It can help determine the buoyancy and stability of a vessel or platform, as well as the amount of weight that can be safely loaded onto it. It is also crucial for calculating the displacement of a ship, which is necessary for navigation and safety purposes.

3. How is the weight of a buoy in water calculated?

The weight of a buoy in water is calculated using the buoyancy equation: W = V x ρw x g, where W is the weight of the buoy, V is the volume of the buoy, ρw is the density of water, and g is the acceleration due to gravity. This equation takes into account the displacement of water by the buoy and the buoyant force acting on it.

4. What factors can affect the weight of a buoy in water?

The weight of a buoy in water can be affected by various factors, such as the density and volume of the buoy, as well as the density and temperature of the water. Changes in these factors can alter the buoyancy force and therefore impact the weight of the buoy in water.

5. How is the buoy in water differential equation applied in real-life situations?

The buoy in water differential equation is commonly used in engineering and naval architecture to design and analyze vessels and other structures that are in or interact with water. It is also used in marine research to study the behavior of marine animals and the effects of buoyancy on their movements.

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