What is the period of a bobbing cork?

[alexander_i]

Homework Statement

A cork floats in water. The cork is a cylinder with radius 1 cm and height 3.4 cm. The density of the cork is 0.55 g/cc. Calculate the period of oscillation if the cork is pushed down a little and released.

Homework Equations

I think I need help with the restoring force... please.

The Attempt at a Solution

-1st I found the mass of the cork - M=D*V =
D = (.55g/cm³)*(1kg/1000g)*(100cm/m)³ = 550kg/m³
V = pi*r²*h = pi*(.01m)²*.034m = 1.068E-5 m³

M = 550kg/m³*1.068E-5m³ = 5.874E-3 kg

-Then setting up my differential:
F_net = F_restore + F_gravity

F_r = -D_h2o*V_h2o : (Density of water * volume of water displaced)

F_g = mg

ma = mg - D_h2o*V_h2o

volume is dependent on y, or the height, so V = pi*r²*y

-rearranging the equation my'' + D_h2o*pi*r²*y = mg

divide by m --> y'' + D_h2o*pi*r²*y/m = g

and setting y=e^rt

r₁=+isqrt(D_h2o*pi*r²*/m)
r₂=-isqrt(D_h2o*pi*r²*/m)

I don't need the particular solution because we need to calculate the period, and

y(t) = Acos{sqrt(D_h2o*pi*r²*/m)t}
+ Bsin{sqrt(D_h2o*pi*r²*/m)t}

the period should be 2*pi/(sqrt(D_h2o*pi*r²*/m) right?

I got .859s but this is not correct. If anyone has some advice, I would be much obliged.

 

[turin]

I would first suggest to redefine y so that y=0 when the cork is at equilibrium. Then, you a have a VERY popular (in physics) type of 2nd order diff. eq., from which you can directly read off the frequency. Actually, it is not really necessary to redefine y, but I think it might make the equation easier for you to recognize.

BTW, y=e^rt is certainly not correct.

 

[RoyalCat]

I would first suggest to redefine y so that y=0 when the cork is at equilibrium. Then, you a have a VERY popular (in physics) type of 2nd order diff. eq., from which you can directly read off the frequency. Actually, it is not really necessary to redefine y, but I think it might make the equation easier for you to recognize.

BTW, y=e^rt is certainly not correct.

Try phrasing the restoring force as a force of the form $$\vec F(y) = -C\vec y$$, and think about what that says about the system.
A good way to start is to look at what the net force on the cork is at equilibrium, and what it is when you've displaced it by a height of $$y$$ into the water (Remember that the force $$mg$$ doesn't change).

A good mental analogy to make in this case would be comparing it to a vertical spring, I assume you've already dealt with that problem, try and remember how you dealt with the effect of gravity there, it's very similar here.