Circular Motion Problem: Downy Ball in Rotating Washing Machine

[irvine752]

Homework Statement

A Downy ball which can be modeled as a sealed, spherical shell of diameter d is rotating with constant angular velocity in a clothes washing machine. Assume that the Downy ball is confined circular path that undergoes no rotations in a plane perpendicular to the angular momentum vector. The sphere is nearly filled with a fluid having uniform densityρ, and also contains one small bubble of air at atmospheric pressure. Your answer should be a function of variables and constants. Assume that the diameter of the sphere is d and the radius of motion is R.
a)The bubble has an initial position directly above the center of the sphere. Where is the bubble, relative to its original position, after the washing machine starts to spin? (Is it closer to the center of the washing machine or further away)

Homework Equations

P = Po + DVh

The Attempt at a Solution

Honestly, I'm not quite sure how to set up this problem. I need help.

 

[berkeman]

What the heck is a Downey Ball, and how does it move in the dryer and in this question specifically? Is it a ball that is pulled part-way up the cylinder of the dryer by the rotation of the cylinder? And what is meant by the "center of the dryer"? Does that mean the axis of rotation of the dryer drum, or the left-right position of the horizontal center of the dryer?

 

[m2003]

I would approach this problem by first identifying the relevant equations and variables. In this case, we are dealing with circular motion, so we can use the equation for centripetal force:

F = mv^2/R

where F is the force, m is the mass, v is the velocity, and R is the radius of the circular path.

Next, we need to consider the forces acting on the bubble in the Downy ball. There are two forces at play: the force due to the rotation of the washing machine and the force due to the buoyancy of the bubble.

The force due to the rotation of the washing machine can be calculated using the equation:

F = mω^2R

where m is the mass of the bubble, ω is the angular velocity, and R is the radius of the circular path.

The force due to buoyancy can be calculated using Archimedes' principle:

F = ρVg

where ρ is the density of the fluid, V is the volume of the bubble, and g is the acceleration due to gravity.

Now, we can set up an equation to find the position of the bubble relative to its original position:

F = mv^2/R + mω^2R - ρVg

We can rearrange this equation to solve for the radius of the bubble's new position:

R = (mv^2 + mω^2R^2 - ρVgR)/F

We know that the initial position of the bubble is directly above the center of the sphere, so we can assume that the initial radius is R = d/2. We also know that the washing machine is rotating with constant angular velocity, so we can assume that ω is constant. Additionally, the volume of the bubble is a function of its initial position, so we can write V = V(d/2). Finally, we can assume that the mass of the bubble and the density of the fluid are constant.

Substituting these values into the equation, we get:

R = (mω^2d^2/4 + mω^2R^2 - ρV(d/2)g)/F

Now, we can simplify this equation by substituting the value for F:

R = (mω^2d^2/4 + mω^2R^2 - ρV(d/2)g)/(mω^2R