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Roll the balls down a shallow ramp: the hollow ball will roll slower!
Why: The answer is Conservation of Energy.
At rest at the top of the ramp, the only energy your ball posses is gravitational potential energy (GPE) - the energy due to gravity. Near the surface of the Earth, this is equal to the mass of the ball (m), times it's height (h) times "g": the gravity constant equal to 9.8 m/s (GPE=mgh). As it rolls down the ramp picking up speed, the ball converts GPE into kinetic energy (Called T for historical reasons) - the energy of motion, equal to 1/2 m v^2 (where v is the speed of the ball).
But, if this is the case, the amount of energy going from GPE to T will be the same for both balls: so they will have the same speed at the bottom of the ramp!
The problem is, we've forgotten an important type of energy: Rotational Kinetic Energy (RKE)! If the balls were sliding with out rolling, the above would be correct, but in order to make the balls roll, you need to give it RKE : this is the energy that makes the ball spin.
RKE has a formula similar to T
RKE=1/2 I w^2
Where, w is the angular velocity (proportional to the frequency with which the ball rolls a full revolution) and I is a quantity called the "Moment of Inertia". If you take all the products formed by multiplying the mass of each little element of the ball by the square of distance of those points from the axis of rotation, and you sum these quantities, you will have the moment of inertia.
By having all it's mass concentrated near the surface of the sphere, the mass elements of the hollow sphere are generally at a larger distance from the axis that corresponding elements of the solid sphere. This means the moment of inertia of the hollow sphere is higher than that of the solid sphere.
When the balls roll down the ramp, the GPE is still used up to create kinetic energies, but now we need both T and RKE. As the hollow sphere has a higher I, more of the GPE has to be converted into RKE, with less energy left over to provide a high T, so the solid sphere rolls faster! from: http://physics.about.com/cs/puzzles/a/070603.htm |
