What Causes the Net Force on a Displaced Ship in SHM?

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The discussion revolves around the net force acting on a displaced ship in simple harmonic motion (SHM). When the ship is displaced deeper into the water by a depth y, the buoyant force changes, leading to a net force calculation. The net force is derived as F = -dAy, where d is the water density, A is the cross-sectional area, and y is the displacement. The weight of the ship does not factor into this equation because the buoyancy force initially equals the ship's weight when floating at equilibrium. Understanding the balance of forces is crucial for analyzing the ship's motion in SHM.
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Homework Statement


There is a ship of mass M floating on a river. The upper portion of the ship has a uniform cross-sectional area A and the density of water is d. Now the ship is displaced a further depth y into the water and released.
What is the resultant force acting on the ship when it is displaced a further depth y into the water?

The Attempt at a Solution


d = m / v where m is the mass of water and v is volume of water being pushed by ship
m = d (A*y)
net F= Mg - mg
net F= g [ M - d (A*y) ]The correct answer is F= - dgAy. Why isn't the weight of the ship taken into account?
Thank you so much!
 
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The ship is floating on the river. So the buoyancy force for the part of the ship that was below the waterline when y = 0 equals its weight Mg.
 
Initially, F_b=dAy_o where y_o is the intial depth of the ship. The depth of the ship when its pushed down by a length dy (or y) is y_o+dy (or y_o+y)
 
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