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Uniform ball rolling without slipping problem

  1. Jan 15, 2009 #1
    1. The problem statement, all variables and given/known data
    A uniform ball of mass M and raduis a can roll without slipping on the rough outer surface of a fixed sphere of raduis b and centre O. Initially the ball is at rest at the highest point of the phere when it is slightly disturbed . Find the speed of the center the G of the ball in terms of the variable theta , the angle between the line OG and the upward vertical. [Assume planar motion]. Show that the ball will leave the sphere when cos (theta)=10/17


    2. Relevant equations

    linear momentum principle

    M*dV/dt=dP/dt=F

    3. The attempt at a solution

    answer: v^2=(10/7)*g(a+b)(1-cos(theta)

    since they give you v^2 and v^2 is associated with the kinetic energy of a particle, should I apply the Energy principle rather than the linear momemtum principle.

    I probably should break the x and y components of the ball with radius b and the hemisphere with to sin(theta) and cos(theta) components. the y- componet will contained the weight of the balls while the x-component will not.
     
  2. jcsd
  3. Jan 15, 2009 #2

    tiny-tim

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    Hi pentazoid! :smile:
    Yes! You must use energy to find the speed …

    how could you apply linear momentum when there's no linear motion and there's gravity?

    But you will need Newton's second law to find when the normal force is zero.
    hmm … probably easier to use radial and tangential components …

    the ball will lose contact when the normal (ie radial) force is zero. :wink:
     
  4. Jan 15, 2009 #3

    V=Mgh = Mg(b+a)cos(theta) (h=0 at theta=90 degrees)

    E=T+V

    at top , ball is at rest
    E(initial)=1/2*M*(0)^2+1/2*I*(0)+Mg(a+b)=Mg(a+b)

    E(final)= 1/2*Mv^2+1/2*(2/5*Ma^2)(v/a)^2+ Mg(b+a)cos(theta)

    E(i)=E(f) ==> Mg(b+a)=1/2*M*(7/5)v^2+Mg(b+a)cos(theta)

    7/10*v^2==g(b+a)(1-cos(theta))

    v^2=10/7* g(b+a)(1-cos(theta))

    not sure how to find the angle between the line OG and the up ward vertical. How would I show that cos(theta)=10/17 when ball leaves sphere?

    to get the normal force would I differentiate v

    m*dv/dt=dP/dt=F=0

    and I can now find theta?
     
  5. Jan 16, 2009 #4

    tiny-tim

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    Hi pentazoid! :smile:

    (have a theta: θ and a squared: ² :wink:)
    Very good! :biggrin:

    (except most people use U for KE, since V looks too much like v :wink:)
    Nooo … as I said, use Newton's second law …

    what are the forces on the ball? …

    they have to equal the centripetal acceleration …

    so work out when the normal force becomes zero. :smile:
     
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