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As you walk through a deep forest, you come upon a clearing. Crossing the clearing, you hear the sound of something landing lightly behind you. Knowing this wood is full of perils, you spin around ready for anything, to find yourself face to face with a Sphinx- not a inanimate statue, but a very much animate man/lion hybrid.
Your attention is drawn immediately to its wicked looking claws and sharp teeth. To your surprise, rather than pouncing, it speaks Human, I am the sphinx. Although I wish to eat you, I am bound by my oath to test you with a riddle before I can feast If you can answer my question, I must let you go free.
Relieved you remember the riddle of the sphinx and prepare your answer about the three ages of man.
Here is my riddle: Beside you are two balls.
"One is hollow, the other is solid. Tell me which is which."
"But... but..."you stammer.
"You were expecting the three ages of man riddle, weren't you? I haven't used that old chestnut for centuries, really." He shakes his head.
"So, I have to tell you which ball is hollow?" You turn and pick up the balls, hoping the hollow one is lighter. Unsurprisingly, the sphinx is too smart for such a simple solution - the balls are of equal weight.
As if it reads your thoughts, the sphinx laughs. The hollow ball is weighted inside with metal, so they are both exactly the same weight.
"I tell you what, I'm not starving, so I'll give you a bit more of a chance. you can use this plank of wood to help you determine the answer. Take your time and think about it, but I warn you, I'll get hungry around lunch time. Then I will not be so generous."Make a shallow ramp out of the plank. Roll the balls down the plank 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 possesses 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 its height (h) times g: the gravity constant equal to 9.8 m/s2
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 without rolling, the above would be correct, but in order to make the balls roll, you need to give RKE, this is the energy that makes the ball spin.
RKE has a formula similar to T, so RKE=½I2
By having all its 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!