Physics Challenge?

Question

The shape of the surface of water in a bucket is determined by the condition that the potential energy per unit mass, as determined in the frame in which the water is at rest, is constant at all places on the surface. Assume that the bucket is into rotation at an angular speed w about its vertical central axis and that the water rotates with the bucket. Write a formula that gives the shape of the water surface. (Hint provided by the book: Note that the normal force of the liquid must account for the centripetal acceleration).

Answer 1

Consider small volume of the liquid near the surface. Pressure in fluid substances in stationary conditions are directed perpendicular to the surface (otherwise we would have unbalanced shear force that would accelerate liquid and motion is not stationary anymore). We have 2 forces acting on our volume: pressure perpendicular to surface and downward gravity. Their sum should provide horizontal centripetal acceleration m*w^2*r. We have right triangle with angle tan(A) = mg/(m*w^2*r) = g/(w^2*r). Then the angle of tangential plane with horizon it tan(B) = 1/tan(A) = (w^2/g)*r. For our curve we have y' = (w^2/g)r, so y(r) = y(0) + (w^2/2g)r^2.

Answer 2

Continuity equation:
1/r * ∂(r*v_r)/∂r = 0
implies that r*v_r = constant
Since v_r = 0 on the wall, v_r = 0 everywhere.

Also there is no variation in the θ direction.

Momentum equation:
-ρ v_θ^2/r = -∂P/∂r
v_θ^2 / r = rω^2 = 1/ρ * ∂P/∂r
v_r = 0

Based on the rotational flow,
v_θ = r*ω
so ∂P/∂r = ρ ω^2 r
P(r) = 1/2 ρ ω^2 r^2 + P0
We already assume no variations inthe θ direction so P0 is a constant not a function of θ

From the general energy equation:
P/ρ + v^2 / 2 + gh = C
h = C - P0/ρg - ω^2 r^2 / g
h = h0 - (ω^2 / g) * r^2

So it's a quadratic relationship where h_0 is the height of the center.

Answer 3

I paintings in something that could probably become a quantum computing gadget. At this 2nd, this is all at a examine point; uncertain no remember if or no longer they're going to become operational every time quickly.

Answer 4

It is a parabola.

The centipetal force will vary with r^2 and the gravitational force, which is perpendicular, will be constant.

Hope this helps.