Mechanical Vibrations Coursework: Frequency of Block Rolling in Water

[tonykoh1116]

Homework Statement

I am currently taking mechanical vibration course and am given coursework.

A wooden block (20X200mm) floats half submerged in water. Determine the frequency of small oscillations of the block rolling from side to side. In this motion, the centre of mass remains in the plane of the water surface.

This is the question with the picture and nothing else is given.
I do not even know where to start. It ONLY gives the size of the wooden box and asks to figure out the frequency...
Only thought I had was that this could be seen as a torsional vibration(shaft-disk vibration) since it shows torsional motion. but that was it, I was not able to make a progress further.

How do I approach this question?
Thank you in advance.

Homework Equations

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The Attempt at a Solution

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[rude man]

EDIT: changed my mind, I guess it really is a torsional problem. Not that it changes anything:
Write the equation relating torque to angular acceleration (similar to F=ma). What is the torque on the block about the rotation axis for a small angular deflection?

 

[tonykoh1116]

EDIT: changed my mind, I guess it really is a torsional problem. Not that it changes anything:
Write the equation relating torque to angular acceleration (similar to F=ma). What is the torque on the block about the rotation axis for a small angular deflection?

Στ = Iα (of course, tau and alpha are vector quantities)
so I (moment of inertia of the block) would be 1/12*m*(a^2+b^2) and α would be double dot of θ.
but I am slightly confused with how to deal with torque part. I know the force acting on the block is the pressure due to buoyant force and the difference in pressure on the left and right side of the block causes the block to rotate.

 

[rude man]

Στ = Iα (of course, tau and alpha are vector quantities)
so I (moment of inertia of the block) would be 1/12*m*(a^2+b^2) and α would be double dot of θ.
but I am slightly confused with how to deal with torque part. I know the force acting on the block is the pressure due to buoyant force and the difference in pressure on the left and right side of the block causes the block to rotate.

That's right.
I'll have to take your word for the rotational inertia of the block; haven't computed it myself.

But OK, the remaining part is to determine the effect on torque of the difference in buoyancy between the left and right sides as you say. If you make the oscillation angle small you can come up relatively easily with an expression for the net torque as a function of the tip angle.
Since torque = force times lever arm you'll have to perform some kind of integration of the buoyancy force with distance from the block's center.

 

[SteamKing]

Στ = Iα (of course, tau and alpha are vector quantities)
so I (moment of inertia of the block) would be 1/12*m*(a^2+b^2) and α would be double dot of θ.
but I am slightly confused with how to deal with torque part. I know the force acting on the block is the pressure due to buoyant force and the difference in pressure on the left and right side of the block causes the block to rotate.

When the block rotates from its original floating position, one side becomes submerged and the other side comes out of the water slightly. In doing so, the center of buoyancy of the block (i.e., the centroid of the displaced volume of water) shifts toward the side which is submerged deeper. The center of gravity of the block doesn't shift, so there is a righting moment which develops for each angle θ that the block rotates. The righting moment is the product of the displacement (or weight) of the block multiplied by the lateral separation of the line of action thru the centers of gravity and buoyancy of the block.

This diagram illustrates what happens when a vessel (or a wooden block) heels to one side:

​

In the figure above, the distance GZ is the moment arm, and B₁ is the center of buoyancy for the heeled vessel.

Since we are dealing with (hopefully) small angles of oscillation, and we are analyzing a wooden block, which has a simple shape, you can approximate the distance GZ by analyzing the triangle GZM. The angle of heel θ is the angle ∠GMZ, so you can calculate GZ if you know the value of GM. G can be determined for the block (it is given in the OP), and the location of the point M (also known as the metacenter) above G can be calculated from the geometry of the block:

http://en.wikipedia.org/wiki/Metacentric_height