Here is a nice problem dealing with physics of chapter 14, and 15: A cylindrical

Question

Here is a nice problem dealing with physics of chapter 14, and 15: A cylindrical cork of mass m, radius r, and height h is floating in a tub of water as shown. The density of water is ?water. You push down on the cork a distance s at it starts bobbing up and down. What is the period of oscillation? State your answer in terms of givens: m, r, h, s, ?water and g. Like always check for reasonableness of your solution. This stuff is new for many of you (it is even new for many instructor), so you should plug in some numbers for a body you have seen bobbing around in water and make sure that your solution is reasonable. If you have never seen a body oscillating in water, build your own experiment and check it out. Have fun with it!

Explanation / Answer

let
So = equilibrium height

At equilibrium:
?(pi r^2)So - m g = 0 -------------By Archimedes Principle

Now perturb it by S:
?(pi r^2)(So + S) - m g = m S'' --------By Newton's 2nd Law, S'' means 2nd derivative.

Combine the last two equations:
=>?(pi r^2)S = m S''

Rearrange previous equation:
=>S'' = [?(pi r^2)/m] S


Solve the above differential equation. Solutions are A sin(wt) + B cos(wt). Where w^2 = ?(pi r^2)/m.

OR equivalently, recall that this is an equation for simple harmonic motion.
SHM equation has the form: x'' = (constant) x => angular frequency = w = (constant)^(1/2)

w = 2 pi/T => T =2 pi/w

Apply to the context of your problem:
T=2 pi/[?(pi r^2)/m]^(1/2)
T=2 pi [m/?(pi r^2)]^(1/2)

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