The Captain loves this TV show called Peep And The Big Wide World, which is a science show for preschoolers. At the end of every show there’s a few minutes where kids do a simple science experiment.
Most of the time I know what they are doing but there is one episode that I can’t explain and it’s bugging me. The kids do the following two experiments:
1. Take two balls of the same size, but one is heavier than the other (one is plastic, the other wood). Roll them down the same ramp at the same time from the same starting point. The heavier one crosses the finish line first. Why?
2. Take two balls of the same weight, but one is bigger than the other. Roll them down the same ramp from the same starting point. The larger ball goes farther. Why?
TurtleHead 
My thought for #1 is that the heavier ball will apply more pressure to the ramp via gravity, and will thus have better grip/friction than the lighter ball. The lighter ball will partially roll and partially slide and that will cause it to go down slower.
I’ve got some thoughts on the second one as well. An item’s inertia does not have any effect on acceleration due to gravity – every item drops at 9.8 m/s/s regardless of size (discounting air resistance for things like feathers, etc.) Therefore, both balls will accelerate down the ramp at a similar speed, since it’s gravity that’s doing the work. Once they’re rolling along the level ground, gravity isn’t directly in the picture any more. What slows the balls down is loss of energy through friction. Both balls will have the same (miniscule) amount of surface area touching the ground, and will thus be affecting by friction in the same amount. The larger ball has a greater inertia though, and will require more energy to stop it, and therefore it goes farther.
Hmm… or in shorter terms, the larger ball will have more potential energy at the same height than a smaller ball (potential energy = mass X gravitational acceleration X height) and thus it has more energy to make it travel farther.
Well, from what I can remember of my physics, both of these are angular motion problems:
1) The force it takes to get something turning is called Torque. Torque isn’t just dependent on sheer mass…it’s also dependent on the distribution of mass in an object (this is called “Moment of Inertia”). Because the plastic ball is hollow, all it’s mass is concentrated at the edge. It takes a lot more torque to get it rolling than it takes for the wood ball, which has its mass evenly distributed throughout. Since the same force is acting on both balls (gravity), the plastic ball accellerates more slowly, because more force is used up getting it rolling.
2) The bigger ball goes farther because it, again, has a bigger moment of inertia (due to its bigger radius). Both balls travel the same distance to get to the bottom of the ramp, both with the same force being exerted on them: because the bigger ball has a bigger MoI, it has more energy when it gets to the bottom. That energy then has to be taken away by friction to make the ball stop…hence the big ball goes further.
A lot of that sounds completely arbitrary, doesn’t it? Without any math, I could have just made all that up. But I’m pretty sure it’s right, from what I remember…
I agree that items always fall with the same rate of acceleration, but don’t they have a different terminal velocity — the maximum speed that they can go? told me that terminal velocity is based on density, so I’m thinking maybe this is why the wood ball goes faster than the plastic ball — it has a higher terminal velocity. But really I’m just guessing at that :).
As for the second question, both the small ball and the large ball have the same amount of mass (the small ball in this specific case is a glass marble, and the large ball is a plastic ball about the size of a softball). So since they have the same mass, doesn’t that mean they will have the same potential energy at the top of the ramp?
Man, I wish I’d paid more attention in university :).
I definitely buy this “moment of inertia” stuff for the big ball/small ball issue. I can visualise how the bigger ball gets turning with more energy. Sold!
But for the heavy ball/light ball problem, I’m not convinced that the plastic ball accelerates more slowly. As points out above, don’t all objects accelerate at the same rate regardless of their mass? Hmmmm…
Gravity provides a constant acceleration yes, because of the way gravity works. But forces in general are not independent of mass. In fact, the definition of force is:
F = m * a (force equals mass times accelleration).
For example, it takes more gas to get a big heavy car going than a little car, right? By the same token, if you apply the same force to both a light object and a heavy object, then the heavy object will accelerate more slowly.
The same thing is true here, except it works in terms of angular motion.
The torque applied to the object (T) is equal to it’s moment of inertia (I) times its rotational accelleration (a):
T = I * a
So, if the moment of inertia is bigger, the accelleration will be less, because the torque stays the same. It’s exactly the same as in linear motion, except with different letters :).
Terminal velocity isn’t based on the density of the falling object, it’s based on the density of the medium that it’s falling in. When an object is falling, the air friction increases the faster it goes, until the force of air friction pushing up on the object is equal to the force of gravity pulling it down. At that point, there are no more forces causing accelleration, so the object stops falling faster, and just remains at the velocity that it’s going. This speed will be different if something is in water than if it is in air.
The balls have the same potential energy for linear motion, but not the same potential energy for angular motion. An object with a higher moment of inertia contains more “angular potential” :), and hence, more potential energy for rolling.
So since they have the same mass, doesn’t that mean they will have the same potential energy at the top of the ramp?
You’re right, I got the 2 questions mixed up halfway through answering I think, hehe.
sinnick’s probably got the right answers, it’s been awhile since I took Physics, I’m just taking some wild guesses!
Why is it that the math and phyics questions always inspire the most responses? What does that say about who we are?
Anyway, I would break the problems down like this. First, ignore friction and angular momentum/rotational energy, and just think about the energy of linear motion. In both problems, one ball has more gravitational potential energy than the other. For the purposes of calculating GPE, we may assume all balls behave as point particles located at their centers. This is true if the ball is a uniform solid, or if it’s hollow and all the mass is uniformly concentrated at the surface.
So the balls roll down the ramp, and at the end convert their GPE to kinetic energy. With the simple model, in both cases one ball ends up with more kinetic energy than the other. If you equate GPE with KE, you find that mass cancels out and the velocity of KE depends only on the initial height; in fact, it’s proportional to the square root of the initial height. So in problem 2, the larger ball has more speed at the bottom of the ramp because its effective height was greater. Of course at this point we see that our model breaks down, because now it says the balls will roll forever, just at different speeds. However, any reasonable model for friction should suggest that a heavier ball with more speed will go farther than a lighter ball with less speed, all other things (such as size) being equal. So I think that’s a reasonable explanation for problem 2.
For the first problem, the same initial conclusion applies: the speed at the bottom of the ramp depends only on the initial height, which in this case is the same. So in this case, even before we get to the bottom of the ramp, our simple model isn’t good enough. Most likely friction and rotational energy are the main reasons why one ball gets to the bottom faster, but without being able to quantify friction it’s hard to say which matters more.
Sigh… this is such a mathematician’s response – “it’s clear that in principle it is possible to solve the problem… let’s move on”. A real physics geek would have intuition as to whether friction or rotation matters more; I don’t. But it’s an interesting problem – and definitely a university-level problem, not a high-school problem.
Well, thinking about this more, in the second problem the balls actually have the same GPE, so my explanation is simply wrong. The GPE is proportional to the height, but the “height” has to be interpreted correctly. In this case its the difference between the initial height and the final height, and for both balls, that’s just the height of the ramp, and has nothing to do with the radius of the ball. So here again, it seems like it has to do with friction and rotational energy, but I’ll have to think about it some more.