How to Derive the Oscillation Equation for a Conical Float?

[yoyo]

I am trying to derive an equation for the oscillation of a conical float. There is a conical float with a density of delta 2 and its floating on water with density 1. Find the oscillation of the floating conical float. The final solution should be:
d^2y/dt^2=-g+(g((L-y)/L)^3)/p

where y(t) is the height of the conical float over the water, L is the length of the conical float,g is gravity and p is the ratio of the two density delta1/delta2

what i did was i found the submerge volume: (1/3)r^2(L-y)

then the weight of water displaced by the float is:
(1/3)r^2(L-y)*delta2*g

mass of float m=(1/3)r^2(L)*delta2

using this in Newton's second law i got:
d^2y/dt^2=-g+g*p((L-y)/L) where p=delta1/delta2

i am not getting the solution that my professor gave?...can somebody explain where the cubed part came from?

 

[ehild]

I am trying to derive an equation for the oscillation of a conical float. There is a conical float with a density of delta 2 and its floating on water with density 1. Find the oscillation of the floating conical float. The final solution should be:
d^2y/dt^2=-g+(g((L-y)/L)^3)/p

where y(t) is the height of the conical float over the water, L is the length of the conical float,g is gravity and p is the ratio of the two density delta1/delta2

what i did was i found the submerge volume: (1/3)r^2(L-y)

then the weight of water displaced by the float is:
(1/3)r^2(L-y)*delta2*g

use delta1 instead of delta2.

mass of float m=(1/3)r^2(L)*delta2

You can not use the same r for both parts. If R is the radius of the base of the whole conus and r is the same for the submerged part, r/R=(L-y)/L

ehild

 

[ramollari]

You would need at least to show that

$$\frac{d^2y}{dt^2} = ky$$

Also I believe you have to clarify whether the cone is upside down, or with the base down.