Rotating Bucket Dynamics: Oil vs. Water Surface Shapes

[rbwang1225]

Homework Statement

Consider a bucket quarter filled with oil. When the bucket is rotating with the angular velocity ω, derive the equation of the surface h(x), where h is measured from the lowest of the surface and x is measured from the rotational axis. Does it change if we replace oil by water?

Homework Equations

$mω^2x=$centrifugal acceleration

The Attempt at a Solution

I am struggling the force diagram of the curled-up oil surface, but don't have a clear confirmation.

Any help would be appreciated!

 

[gneill]

mω²x would be centrifugal force, not acceleration. Drop the m to get acceleration (f = m*a). Force is just a scaled version of acceleration when the mass stays the same

At steady state You expect a small parcel of fluid at a point on the surface to remain stationary relative to the the fluid around it, so any net acceleration must be directed along the surface normal passing through the parcel and into the body of fluid. That gives you a clue for finding the slope of the surface at any distance from the rotation axis. Then use the fact that df/dx gives the slope of f(x) ...

 

[rbwang1225]

Dear gneill:

Sorry for the typo. Here is my calculation
$\frac{dh}{dx}=\frac{m\omega^2x}{mg}$
Therefore, $h-h_0=\frac{1}{2}\frac{\omega^2}{g}x^2$, where $h$ is the height measured from the surface to the bottom.
Am I right? I am still not clear about what you meant by "any net acceleration must be directed along the surface normal passing through the parcel and into the body of fluid".

Sincerely.

 

[gneill]

Dear gneill:

Sorry for the typo. Here is my calculation
$\frac{dh}{dx}=\frac{m\omega^2x}{mg}$
Therefore, $h-h_0=\frac{1}{2}\frac{\omega^2}{g}x^2$, where $h$ is the height measured from the surface to the bottom.
Am I right? I am still not clear about what you meant by "any net acceleration must be directed along the surface normal passing through the parcel and into the body of fluid".

Sincerely.

Looks okay. The idea I was trying to across is that if the surface of the water is to remain still when it reaches steady state, there can be no further flow. So a small parcel of water at the surface can't experience any net force parallel to the surface -- otherwise it would want to move along the surface. So any net force (acceleration) that the parcel feels must be in a direction along the normal to the surface, and in fact directly into the surface (otherwise the parcel would rise off the surface).

 

[rbwang1225]

I think I got the idea!
Here is my final force diagram of the system.

And therefore, the answer for the second question is that the shape will not change if water is used.

 

[gneill]

I think I got the idea!
Here is my final force diagram of the system.
View attachment 54104
And therefore, the answer for the second question is that the shape will not change if water is used.

Sounds good to me