Buoy in waves problem (Feynman)

[mainguy]

Hi all, would appreciate a spot of help on this problem which originally comes from Feynman's introductory physics course.

1. Homework Statement
'A spar buoy of uniform cross-section floats in a vertical position with a length L submerged when there are no waves on the ocean. What is the amplitude A with respect to the mean ocean surface of the vertical motion of the uoy when there are sinusoidal waves of height h (crest to trough) and period T on the ocean? (Neglect fluid friction)

Homework Equations

I just went for a simple differential equation, first finding an expression from:
mg = Laρg (a = area of buoy, ρ = density of water)
And rarranged to find p = m/LA

The Attempt at a Solution

[/B]
Then from a simple differential equation
mX'' = hApgsin(wt)

Found x = hgT²/L4pi²

Where T is period.

Aaand it was wrong. Vaguely close to the real answer but I'm out by a fair bit, no idea what to correct for as there is no frictional term.

 

[hutchphd]

Hi all, would appreciate a spot of help on this problem which originally comes from Feynman's introductory physics course.

1. Homework Statement
'A spar buoy of uniform cross-section floats in a vertical position with a length L submerged when there are no waves on the ocean. What is the amplitude A with respect to the mean ocean surface of the vertical motion of the uoy when there are sinusoidal waves of height h (crest to trough) and period T on the ocean? (Neglect fluid friction)

Homework Equations

I just went for a simple differential equation, first finding an expression from:
mg = Laρg (a = area of buoy, ρ = density of water)
And rarranged to find p = m/LA

The Attempt at a Solution

Then from a simple differential equation
mX'' = hApgsin(wt)

Found x = hgT²/L4pi²

Where T is period.

Aaand it was wrong. Vaguely close to the real answer but I'm out by a fair bit, no idea what to correct for as there is no frictional term.[/B]

You need to think a little more...what is the differential equation in the absence of waves?...then add the driving force

 

[mainguy]

You need to think a little more...what is the differential equation in the absence of waves?...then add the driving force

In the absence of the wave surely the differential is just the bouyancy take away the weight or:

mx'' = ρLAg - mg

Then when the waves are added L becomes a function of t:

mx'' = ρAg(L + hsin(wt)) - mg

But this just leads down another dead end :/

I feel like I'm missing some fundamental aspect of the physical scenario?

 

[haruspex]

mx'' = ρLAg - mg

No, you need to consider a displacement x from equilibrium (even without waves).

 

[mainguy]

No, you need to consider a displacement x from equilibrium (even without waves).

Thanks, so there is a restoring force proportional to the displacement from equilibrium:

F = -xAρ

Still no idea where to take this

 

[haruspex]

Thanks, so there is a restoring force proportional to the displacement from equilibrium:

F = -xAρ

Still no idea where to take this

You missed the factor g.

You now need to be careful how you define your variables. I would keep x as the displacement from the equilibrium position in flat water, i.e. from a fixed point in space. So you need to express the excess buoyancy at time t, taking the waves into account.

 

[hutchphd]

Thanks, so there is a restoring force proportional to the displacement from equilibrium:

F = -xAρ

Still no idea where to take this

Being careful about what you choose to call x=0 put all these net forces into the Newton equation for motion with the wave.. ( Don't forget gravity!) You should recognize the result as an inhomogeneous second order differential eqn (have you seen these?) specifically called a driven harmonic oscillator. This equation is ubiquitous and you should master it inside and out... Incidently this is a very nice problem...

 

[mainguy]

You missed the factor g.

You now need to be careful how you define your variables. I would keep x as the displacement from the equilibrium position in flat water, i.e. from a fixed point in space. So you need to express the excess buoyancy at time t, taking the waves into account.

Ah my bad, haha, sorry it was late.

So then the equation looks fairly similar, as x is surely just hsinwt?

So F = - hAρgsin(wt)

If we integrate this a few times w

X = C - hρgsin(wt)/Lω²

X = C - (hgT²/4π²)sin(wt)

But C is 0 as at t = 0 x = 0

So amplitude is just the part out front of sin, right?

Feynman's answer has this part and an additional constant, however I can't see why the constant wouldn't disappear as surely the rod won't oscillate about some new equilibrium :/ :'(

 

[hutchphd]

So then the equation looks fairly similar, as x is surely just hsinwt?

Exactly what is the equation from Newton's law? Write it down.

 

[mainguy]

Exactly what is the equation from Newton's law? Write it down.

Oh ok, is it:
mx'' = mg - buoyant froce

And the buoyant force varies according to sine?

 

[hutchphd]

Oh ok, is it:
mx'' = mg - buoyant froce

And the buoyant force varies according to sine?

Yes the buoyant force will depend upon the difference between the position of the buoy and that of the surface. So write it down (with care about what x=0 is)

 

[mainguy]

Yes the buoyant force will depend upon the difference between the position of the buoy and that of the surface. So write it down (with care about what x=0 is)

Thanks a bunch for guiding me through this, it's really helpful.

So I thought the buoyant force is, taking x as the distance from the starting position where F_b = mg

(L + h - x)*Aρg
Where h = h₀sinwt

mx'' = mg - LAρg - hsinwtAρg + xAρg

Then the first two terms cancel as they're equal in equilibrium:

mx'' = xAρg - hsinwtAρg

I feel like I'd need to apply complex numbers/exponential guesses to get this? Still new to differential eqs :/

 

[haruspex]

mx'' = xAρg - hsinwtAρg

Careful with signs. Which way are you taking as positive for x? It should be evident that changes in x lead to an opposing acceleration, not a reinforcing one.

There's plenty online on forced oscillations, e.g. https://math.libretexts.org/TextMap...r_ODEs/2.6:_Forced_Oscillations_and_Resonance.
All those I found also allow for damping, but you can just discard that part.

 

[hutchphd]

Very good but check your signs. Soon you will know many ways to solve this. The good news is that all the ways are kosher including guessing (as long as it solves every part of the problem). Since you intuit that in steady state this thing will be oscillating with the incident wave try x=constant sinwt and see if you can solve for the constant. Notice the sharp (infinite with no friction) response at the natural "bobbing" frequency of the buoy. If you really understand this system you will understand about 25% of physics... maybe more! Good luck..