In reference to a previous question
hear to read
In this case we have the same closed, rotating
hollow cone whose rotation at some tangential
velocity w is perpendicular to the direction of
gravity (so the tip of the cone is pointing down,
and the cone rotates on a horizontal plane). The
incline of the cone is 45 degrees, but I suppose
it can be any angle x. The cone has a radius r.
Instead of containing several spheres, like my
last question, this cone contains some fluid of
known volume, density, and mass. As the cone
begins to rotate, the fluid co-rotates with it,
climbs up the incline and presses up against the
top lid. Eventually the cone's tangential velocity
is stabilized and the fluid exerts a constant
force on the inside of the cone.
Knowing this information, how does the physics
in calculating the x and y force components of the
force that the fluid exerts on the inside of the
cone change from the case of the rotating spheres?
How can one calculate the x and y force components
of the force that the fluid exerts on the incline
and lid of the cone, respectively?
Thank you for your help!
Hmm, this problem is trickier than the one with
rotating spheres. The physics is essentially the
same as for the rotating sphere case, but the
complication is that fluids don't behave like hard
objects. In particular, when we think of a fluid
exerting a force, we think of the fluid's pressure
as the thing responsible for what's exerting the
force (as opposed to the normal force, in the case
of the spheres). Unlike the normal force for the
sphere, which is a force acting only at a point,
the fluid pressure acts everywhere the fluid is
touching the walls of the cone, and in particular
can vary from place to place. In order to
understand how this pressure varies, you need two
things: an equation of state for the fluid, which
tells you how the density of the fluid is related
to its temperature and pressure, and the equation
of hydrostatic equilibrium, which tells you how
the pressure must change as a function of position
and applied forces (e.g. gravity, the centrifugal
force). For a simple case, you can consider an
incompressible fluid, so that the density is
constant everywhere. Then you just need to worry
about the equation of hydrostatic equilibrium to
understand how the pressure varies throughout the
fluid as a function of the speed at which you're
rotating the cylinder.
This is a tricky (but interesting!) problem,
and I'm not sure offhand how easy it is to do.
There may be shortcuts that make it simpler, but I
can't think of any offhand. If I do think of any,
I'll let you know.
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