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Spherical Shell with Fluid

  1. Jan 19, 2015 #1
    1. The problem statement, all variables and given/known data
    A spherical shell of mass M and radius R is completely filled with a frictionless fluid, also of mass M. It is released
    from rest, and then it rolls without slipping down an incline that makes an angle θ with the horizontal. What will
    be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is g.
    The moment of inertia of a thin shell of radius r and mass m about the center of mass is I =2/ 3mr2; the moment
    of inertia of a solid sphere of radius r and mass m about the center of mass is I =2/5mr2.

    (A) a = g sin θ
    (B) a =3/4 g sin θ
    (C) a =1/2 g sin θ
    (D) a =3/8 g sin θ
    (E) a =3/5 g sin θ


    .

    2. Relevant equations


    3. The attempt at a solution
    I considered the fluid inside to be a "sphere in sphere" and used conservation of energy to get 2M*g*h=M*v^2 + 2/5*Mv^2+ 2/3 * Mv^2. (I used the fact that 2*1/2 =1 and that v=omega*r to simply). Adding the right hand side and diving by M give me 2*g*h= 29/15 v^2. I then divide by 2*h to get g=(29/15*v^2)/2h. How do I go from here?
     
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  3. Jan 19, 2015 #2

    haruspex

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    How will a fluid filled sphere behave differently from a solid sphere of the same mass distribution? (Not a rhetorical question.)
     
  4. Jan 19, 2015 #3
    Won't the density inside the the sphere be less than the exterior since there is the same mass with a far larger volume in the interior?
     
  5. Jan 19, 2015 #4

    haruspex

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    I wrote "same mass distribution'. I.e., what if the fluid were replaced by an equally dense solid, not the same as the sphere's shell, but fixed rigidly inside it?
     
  6. Jan 19, 2015 #5
    I would say then there would be no difference.
     
  7. Jan 19, 2015 #6

    haruspex

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    No, there's a difference. Imagine a large basin filled with water in front of you. You seize the handles on the sides and turn it suddenly. Does it require a lot of torque?
     
  8. Jan 19, 2015 #7
  9. Jan 19, 2015 #8

    haruspex

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    Well, no it doesn't. Nowhere near as much as if it contained the same mass of ice. Can you not see why?
    Fill a cup with water and sprinkle something on top that you can then observe floating. Rotate the cup. What do you notice? What would look different if the water were solid?
     
  10. Jan 19, 2015 #9
    I noticed that the floating thing is rotating as well and I think that if the water was solid that nothing would rotate.
     
  11. Jan 19, 2015 #10

    haruspex

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    relative to the room or relative to the cup? (I mean, while the cup is rotating.)
     
  12. Jan 19, 2015 #11
    Relative to the cup.
     
  13. Jan 19, 2015 #12

    haruspex

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    Right, the liquid does not rotate fully with the cup. It does rotate a bit because of drag between the cup and water and within the water, but go back and read the problem statement. What does it tell you about the fluid here?
     
  14. Jan 19, 2015 #13
    It's frictionless.
     
  15. Jan 19, 2015 #14

    Nathanael

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    If the liquid in your cup was frictionless, would it rotate (relative to the room) when you rotated the cup?

    What about the liquid inside the rotating sphere?
     
  16. Jan 19, 2015 #15
    Yes, so same with the sphere.
     
  17. Jan 19, 2015 #16

    Nathanael

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    It would rotate relative to the room if it were frictionless? If there's no friction between the cup and the liquid how does it get rotating?
     
  18. Jan 19, 2015 #17
    Well I thought that if it was frictionless that the it wouldn't rotate with respect to the cup and that since the cup is rotation with respect to the room that the liquid would also be in rotation relative to the room.
     
  19. Jan 19, 2015 #18

    Nathanael

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    By "it wouldn't rotate relative to the cup" you essentially mean "it would rotate with the cup" right?

    The liquid has inertia, it takes some force (friction) to get it to rotate with the cup. Otherwise, it would rather sit still.
     
  20. Jan 19, 2015 #19
    Man all the reference frames can get confusing sometimes, but I'm starting to get what you're saying.
     
  21. Jan 19, 2015 #20

    Nathanael

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    Consider a bowl of soup. If you rotate the bowl slowly, the soup rotates with it.
    But what happens when you rotate the bowl quickly? The soup still rotates with the bowl slightly, but not nearly as much.
    The faster you rotate the bowl, the less the soup rotates.

    This is because rotating the bowl faster and faster is like reducing the force of friction. You're not actually reducing the force of friction, but you are reducing the time that the friction is acting for (thereby reducing the effect of friction). So you see, the more you reduce the friction, the less the soup rotates. If the soup was somehow frictionless, it would not rotate at all!

    So back to the ball filled with liquid rolling down the hill (an interesting problem!).
    It will be like a spherical shell rolling down a hill with a ball of liquid just "floating" down the hill (i.e. not rotating). See if you can't work out the mathematics of that situation.
     
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