Too many pies?

Question

A 60-degrees cone filled with water was brought to rotation
around it vertical axis with period of rataion 2s and then stopped.
The water remaing inside the cone was poured into cylindrical well
of diameter 1m and the depth of water in the well turned out to be 3m.

What is the value of accelaration g?
(answer in closed form, please)

60-degrees cone is formed by rotating equilateral triangle aroung its axis of symmetry.

"where ω = 2"

Nope

Answer 1

The profile of the cavity is parabolic of the equation:

h(r) = (ω r)² / 2g

The parabolic cavity thus has a volume of

(π ω² r^4) / 4g

The cone has a volume of

(π/√3) r³

so that the volume of the water is

(π/√3) r³ - (π ω² r^4) / 4g = (3/4) π

where ω = 2. But of course, the missing piece of the puzzle, not stated, is that it's assumed that all excess water sweeps out of an open top cone, so that the parabola formed by the water is tangent to the wall of the cone. This is where the big if is. Thus, we have

R = (√3 g) / ω²

so that plugging this in, we solve for g and get

g = 4 m/sec²

I give this one a star for having first made me think this was indeterminate.

Addendum: If ω = 0.5, then g = 0.25 m/sec² Are we on an asteriod?

Answer 2

MMM pies, yummy.

I'll answer your question later maybe.

Couldn't resist adding to your misery by giving an extra dose of lame answers.

:)

I'll get you started though.

The top surface of the water in the spinning cone has to be a surface of equal energy.

potential + kinetic energy is constant.

mgh + 1/2 mv^2 is constant
so
gh + 1/2 (v^2) is constant
where
v = (omega)radius
so
gh + 1/2 (omega)(radius) = constant

At the lip of the cone

g(height of cone) + 1/2 (omega)(radius of come) = constant

That will let you solve for the constant as a function of (g) and the cone size.

Of course the geometry of the cone tells you that the radius of the cone is just the height over sqrt (3).

Now you can get an equation for the equal energy surface.

Use some calculus to find the volume of water in the cone underneath the surface. It will be a function of g and the geometry of the cone.

Then set that volume equal to the volume you get in the pour-out well.

Solve for g.

Nasty problem, but doable if they give you the size of the cone (which I don't see that they did). Otherwise, your answer for g will end up being expressed in terms of the cone size.

Answer 3

Pies = yummy.
Me <3 pies.
Especially chicken pies.
And game pies.
And fruit pies.

Answer 4

too many pies would make you as fat as a little piggy.