Application to physics
A cylinder whose cross-section is an isolceles triangle with sides 5 in, 5 in, and 6 in has a height H = 20 in and weight 5 Ib. After being pushed down so that its top is level with the surfaceof the water and released, the cylinder bobs, with axis vertical, in water whose weight per unit volume is 62.5 Ib/ft^3. Solve for the motion y(t) where y(t) is the position relative to equilibrium. Determine, and state clearly, the period and amplitude of the motion. HINT: Draw a picture.
I really don't have a clue on this one, so help would be greatly appreciated. Thanks!
I don't really like this question. I'll tell you why later.
Originally Posted by SoBeautiful
But first, the unit is "lb" NOT "Ib."
You know that the upward (bouyant) force on the cylinder is equal to the weight of the displaced water. The weight of the displaced water is equal to the volume of the cylinder that's under water at time t, times the density of the water.
So start with equilibrium. The density of the cylinder is (presumably) less than the density of the water, so the cylinder will float. You need to find how much of the cylinder is under water and you can do that by a simple Newton's 2nd Law problem. (Fnet = B - w = 0 where B is the bouyant force and w is the weight of the displaced water. etc.)
Now assume you give the cylinder a SMALL downward displacement. Why small? Because only a small displacement will produce harmonic motion. (If you don't understand why, just look at the first few terms of the Taylor series for a displacement of this kind. The constant term will be missing as long as we take the origin to be the equilibrium height, and if the restoring (bouyant) force is assumed to be a linear function of the height (as it will be in a first approximation, ie. small displacement) then the motion obeys the equation F = my'' = -ky, which is the harmonic oscillator equation.)
The difficulty with this programme is that as the cylinder sinks, the water level rises, so the equilibrium level shifts. I suppose we can assume an infinite reservoir of water for the purposes of solving the problem so that the water level never technically rises, but I hate making that kind of assumption without checking with whoever is asking for a solution.
If the water level never rises then you know what the form of the bouyant force is, and it is going to depend directly on the displacement of the CM of the cylinder from equilibrium. Then set B(y) = My'', where M is the mass of the cylinder.