A SHM problem of floating

[crossfacer]

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!

 

[AlephZero]

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.

 

[chaoseverlasting]

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 $$y_o+dy (or y_o+y)$$