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Problem rotating cilinder

  1. Jan 12, 2008 #1
    1. The problem statement, all variables and given/known data
    A cylinder with radius R is rotating about a fixed axis in a horizontal plane with a constant angular velocity
    [tex]\omega _0[/tex]. A piece of clay is describing a path that is tangent to the rotating cylinder and in a direction opposite to the direction of rotation of the cylinder. When the piece of clay touches the cylinder, it sticks on it. What initial velocity does the piece of clay have to have in order to bring the cylinder to rest?
    I tried to draw the situation:
    [​IMG]

    2. Relevant equations
    Conservation of angular momentum and conservation of linear momentum.


    3. The attempt at a solution
    The goal is to find an expression for v, the speed of the piece of clay before the collision.Since it's an isolated system, there is conservation of angular momentum and conservation of linear momentum. There is no conservation of mechanical energy because it's not an elastic collision. Because the piece of clay sticks on the rotating cylinder after the collision, it is a perfectly inelastic collision.
    I defined the initial situation as just before the piece of clay collides with the rotating cylinder. The angular momentum is then
    [tex]L_i = Rmv-I\omega _0[/tex] with
    [tex]m =[/tex] the mass of the piece of clay
    [tex]v =[/tex] the speed of the piece of clay
    [tex]I =[/tex] the moment of inertia of the cylinder
    We want the system to be at rest in the final situation so the angular momentum is then
    [tex]L_f = 0[/tex]
    Because there is conservation of angular momentum,
    [tex]Rmv-I\omega _0=0[/tex]
    [tex]\Leftrightarrow Rmv = I\omega _0[/tex]
    [tex]\Leftrightarrow v = \frac{I\omega _0}{Rm}[/tex]
    We don't know the mass of the piece of clay m so we must be able to find an expression for v without the mass m. Until here I could do it all right.
    There is also conservation of linear momentum. That's where I'm stuck. Initially, the piece of clay has linear momentum and the rotating cylinder doesn't (it only has angular momentum). So
    [tex]p_i = mv + 0[/tex]
    However, in the final situation we want the linear momentum to be zero because we want the system to be at rest! Thus
    [tex]p_f = 0[/tex]
    But because of conservation of linear momentum,
    [tex]p_i = p_f[/tex]
    [tex]\Leftrightarrow mv = 0[/tex]
    [tex]\Leftrightarrow v=0[/tex]
    But that's not possible. Where did it go wrong? I noticed that I didn't use anywhere that the collision is perfectly inelastic... Can someone help me?

    Thanks,

    Yoran
     
  2. jcsd
  3. Jan 12, 2008 #2

    Doc Al

    User Avatar

    Staff: Mentor

    Linear momentum is not conserved. It's not an isolated system: The cylinder rotates around a fixed axis, which exerts a force on it when it's hit by the clay.

    But angular momentum is conserved. Your thinking there is correct. But there's no way to find v without being told the mass of the clay, so just leave your answer in terms of m. (What information do you have? Did you present the problem exactly as it was given?)
     
  4. Jan 12, 2008 #3
    Ok, thanks a lot! This is exactly the problem I was given, but I guess it is incomplete then in the sense that there is not enough information given.
    But what to do with the information that the ball sticks to the cylinder? Is that unnecessary information? Or can it be used in any way...
    Anyway thanks a lot for the quick reply!
     
  5. Jan 12, 2008 #4

    Doc Al

    User Avatar

    Staff: Mentor

    You are using that information. If they didn't stick together, how could you deduce that the final speed of the cylinder was zero?
     
  6. Jan 12, 2008 #5
    Damn, sometimes I feel so stupid...
     
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