F = ma 2007 Exam - Question #25 - SHM with a box in water

[physicsod]

Homework Statement

Find the period of small oscillations of a water pogo, which is a stick of mass m in the shape of a box (a rectangular parallelepiped.) The stick has a length L, a width w and a height h and is bobbing up and down in water of density ρ. Assume that the water pogo is oriented such that the length L and width w are horizontal at all times. Hint: The buoyant force on an object is given by Fbuoy = ρVg where V is the volume of the medium displaced by the object and ρ is the density of the medium. Assume that at equilibrium, the pogo is floating.

Homework Equations

Fbuoy = ρVg.
Fg = mg

The Attempt at a Solution

So I created the general box (or "parallelepiped") and set it's dimensions as L, w, and h. I let the height underneath the water be "a," where a is some fraction of h. Then, the buoyant force is Fbuoy = ρLwag. But how do I extract the period from this equation?

 

[Simon Bridge]

The buoyant force is the restoring force.
Your parameter "a" is the bit that has SHM.

 

[physicsod]

Well yeah, I understood that bit. Is there a general way to go from the restoring force to a general SHM equation?

EDIT: I just thought of setting the gravitational potential energy (mga) equal to the work done by the buoyancy force, for conservation of mechanical energy. Is this correct?

 

[Simon Bridge]

Ahhh... the usual way is just to set ƩF=ma ... so it is unfortunate that you have used a as your emmersion parameter.

If h₀ = the equilibrium level, then set y = displacement from equilibrium downwards, thus: \vec{F}_{bouy}+\vec{F}_{grav}=m\ddot{\vec{y}}, resolve the forces in terms of m and y, and solve the differential equation.
Approximate for small a small initial downwards displacement y₀ ... which will become the amplitude of the oscillations.

 

[physicsod]

Yuck, so this is a basic diff eq question? Darn, well I guess that's expected for question 25 on the exam. I'll try that out, thanks. :)

 

[Simon Bridge]

Yeah - though it has a standard solution - like the mass on a spring: buoyancy standing in for the spring force. So, if you understand the physics you can just do it by substitution.

They don't really expect you will solve the DE in an exam - that's just to slow you down if you didn't understand the material. Too little understanding and you won't be able to complete the exam. Sneaky see - ideally they want you to understand what you are doing but if you are really fast at the math, well, that's good too. Most people do a bit of both.