1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Classical mechanics: ball rolling in a hollow sphere

  1. Feb 22, 2008 #1
    [SOLVED] Classical mechanics: ball rolling in a hollow sphere

    1. The problem statement, all variables and given/known data
    This problem is from Gregory:

    A uniform ball of radius a and centre G can roll without slipping on the inside surface of a fixed hollow sphere of (inner) radius b and centre O. The ball undergoes planar motion in a vertical plane through O. Find the energy conservation equation for the ball in terms of the variable [tex]\theta[/tex], the angle between the line OG and the downward vertical. Deduce the period of small oscillations of the ball about the equilibrium position.

    So in summary, we have:
    a: radius of ball
    m: mass of ball
    [tex]\theta[/tex]: angle of the ball's position, relative to the vertical line connecting the center and bottom of the hollow sphere
    I: moment of inertia of ball
    [tex]\omega[/tex]: rotational velocity of ball
    T: kinetic energy of ball
    V: potential energy of ball (V=0 at height [tex]\theta[/tex]=[tex]\pi[/tex]/2, the center of the sphere)
    E: total energy of ball
    g: acceleration due to gravity

    2. Relevant equations
    I = 2/5ma^2

    3. The attempt at a solution
    First of all, I'm assuming that [tex]\omega[/tex]=[tex]\theta[/tex]'. It sounds intuitive, but I could be wrong there.

    I'm given, as a solution, that the period of small oscillation (that is, sin([tex]\theta[/tex])=[tex]\theta[/tex]) is 2[tex]\pi[/tex](7(b-a)/5g)^(1/2), which I'm not getting in my results. I have a very strong hunch that my mistake comes from bad energy equations. So, would you mind taking a look of these?

    T = 1/2mv^2 + 1/2I[tex]\omega[/tex]^2
    v = [tex]\omega[/tex]*(b-a)
    So T = 1/2m([tex]\omega[/tex]*(b-a))^2 + 1/2(2/5ma^2)[tex]\omega[/tex]^2
    T = m[tex]\omega[/tex]^2/10(7a^2-10ab+5b^2)
    V = -(b-a)mgcos([tex]\theta[/tex])

    So E = T + V = that stuff
    Am I correct here?
  2. jcsd
  3. Feb 22, 2008 #2

    Shooting Star

    User Avatar
    Homework Helper

    That is wrong. Draw a simple diagram to figure it out.
  4. Feb 22, 2008 #3
    Hah, it's always the little mistakes in the beginning that steal away an hour of my life.

    That fixed everything. Thank you so much for catching that.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Classical mechanics: ball rolling in a hollow sphere
  1. Rolling ball mechanics (Replies: 4)