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Question about moment of interia

  1. Feb 22, 2010 #1

    objects with different radius got same speed.
    Lecturer says it's because this is independent from radius and mass.

    But the hollow one is slower.
    Lecturer say it's because of moment of interia.

    But moment of interia is radius dependent.

    So in example with different radiuses moment of interia is also different.
    So why speed is the same?
  2. jcsd
  3. Feb 22, 2010 #2
    The rotational kinetic energy of a hoop is 0.5*(mass of the hoop * radius of the hoop^2)*(velocity / radius of the hoop)^2.

    After squaring the (velocity / radius) you see the radius^2 is in both the numerator and denominator, so it's gone. So, we know that the rotational kinetic energy of a hoop with pure roll is 0.5*(mass of the hoop)*(velocity)^2

    A similar method arises in the case of a solid cylinder of uniform mass density, where the moment of inertia is instead (0.5 * mass of the cylinder * radius of the cylinder^2), so the calculations yield 0.25*(mass of the cylinder)*(velocity)^2.

    Using conservation of energy principles, we set the translational kinetic energy of the center of mass + the rotational kinetic energy of the object = the gravitational potential energy at the top of the inclined plane.

    So the mass cancels out of the potential, the translational and rotational kinetic energies of both objects. This leaves us with the velocity of the hoop at the bottom of the plane as the square root of g*h, where h is the length of the plane *the angle at which it is inclined. Similarly, we find the velocity of the cylinder to be the square root of (4/3)*g*h

    So, if the two objects are of the same material, we can draw the conclusion that a solid cylinder will roll faster than a hoop of any size
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