# ODE Oscillations

1. Oct 16, 2013

### djh101

1. A spherical buoy of radius r floats half-submerged in water. If it is depressed slightly, a restoring force equal to the weight of the displaced water presses it upward; and if it is then released, it will bob up and down. Find the period of oscillation if the friction of the water is neglected.

T = 2π/ω
y'' + ω2y = 0
F = my'' = .5Vwaterρwaterg
y = cos(ωt) & y'' = ω2cos(ωt)

So ω needs to be found, which can be done by setting t = 0 and amax = ω2 = .5Vρg.
.5V = 2/3πr3 and ρ = 1, so ω2 = 2/3πr3g.
All that is left is to take the square root and divide from 2 to get T. However, the book gives the answer 2π√(2r/3g)s.
Summary: $2\pi\sqrt{\frac{3}{2\pi r^{3}g}} \neq 2\pi\sqrt{\frac{2r}{3g}}$

2. Suppose that a straight tunnel is drilled through the earth between any two points on the surface. If tracks are laid, then a train placed in the tunnel at one end will roll through the earth under its own weight, stop at the other end, and return. Estimate the value of the time required to complete one round trip.

y'' = g = GM/y2
M = 4/3 πR3ρ
y'' = 4/3 πGρy, which gives ω2 = 4/3 πGρ. The book, however, gives T = 2π√(R/g).
Summary: $2\pi\sqrt{\frac{3}{4πGρ}} \neq 2\pi\sqrt{\frac{R}{g}}$

Am I over thinking these? The answers in the book are pretty simple, but I'm having a little bit of a hard time figuring out where they came from.