Imagine rolling two identical cans down a slope, but one can is empty and one is full. Which one will reach the bottom first? You might have learned that when dropped straight down, all objects fall at the same rate regardless of how heavy they are (neglecting air resistance). Is the same true for objects rolling down a hill? Try this experiment to find out!
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When you lift an object up off the ground, it has potential energy due to gravity. The amount of potential energy depends on the object’s mass, the strength of gravity, and how high it is off the ground. When you drop the object, this potential energy is converted to kinetic energy, or the energy of motion. Kinetic energy depends on an object’s mass and its speed. Ignoring frictional losses, the total amount of energy is conserved.
For a rolling object, kinetic energy is split into two types: translational (moving in a straight line) and rotational (spinning). So, when you roll a ball down a ramp, it has the most potential energy when it is at the top of the ramp, and this potential energy is converted to both translational and rotational kinetic energy as it rolls down the ramp. This leads to the question: will all rolling objects accelerate down the ramp at the same rate, regardless of their mass or diameter?
The answer depends on the objects’ moment of inertia, or a measure of how “spread out” their mass is. If two cylinders have the same mass but different diameters, the one with a bigger diameter will have a bigger moment of inertia, because its mass is more spread out. Similarly, if two cylinders have the same mass and diameter, but one is hollow (so all its mass is concentrated around the outer edge), the hollow one will have a bigger moment of inertia. Does moment of inertia affect how fast an object will roll down a ramp? Try this experiment to discover the surprising result!
Extra: find more round objects (spheres or cylinders) that you can roll down the ramp. For example, rolls of tape, permanent markers, plastic bottles, different types of balls, etc. Try racing different types objects against each other. What seems to be the best predictor of which an object will make it to the bottom of the ramp first?
Extra: try the experiment with cans of different diameters. What happens if you compare two full (or two empty) cans with different diameters? What about an empty small can vs. a full large can or vice versa?
Extra: try racing different combinations of cylinders and spheres against each other (hollow cylinder vs solid sphere, etc).
Observations and Results
You should find that a solid object will always roll down the ramp faster than a hollow object of the same shape (sphere or cylinder) – regardless of their exact mass or diameter. This may come as a surprising or counterintuitive result! A classic physics-textbook version of this problem asks what will happen if you roll two cylinders of the same mass and diameter, one solid and one hollow, down a ramp. The answer is that the solid one will reach the bottom first. In that specific case, it is true that the solid cylinder has a lower moment of inertia than the hollow one (their mass is the same, but the hollow cylinder’s mass is all concentrated around its outer edge so its moment of inertia is higher). But, it is incorrect to say that “the object with a lower moment of inertia will always roll down the ramp faster.” It takes a bit of algebra to prove (see the Hyperphysics link below), but it turns out that the absolute mass and diameter of the cylinder do not matter when calculating how fast it will move down the ramp - only whether it is hollow or solid. So, in this experiment, you will find that a full can of beans rolls down the ramp faster than an empty can – even though it has a higher moment of inertia (it has the same radius, but is much heavier than an empty aluminum can). Applying the same concept shows that two cans of different diameters should roll down the ramp at the same speed, as long as they are both empty or both full. The same principles apply to spheres as well – a solid sphere like a marble should roll faster than a hollow sphere like an air-filled ball, regardless of their respective diameters.
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Ben Finio, PhD, Science Buddies
Science Buddies |
Mass, gravity, moment of inertia, kinetic energy, potential energy, conservation of energy
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