A jar filled with water reaches the bottom of the ramp before an empty identical jar. Many “explanations” that I see for this invoke “moment of inertia.” This is to say that they explain it by saying that a rolling object with its mass evenly distributed has a greater rotational inertia and will begin to accelerate more rapidly than one that has much less mass and the mass is mainly at the outer limits of the object. Or something. My point is that such “explanations” are not explanations at all - they simply say the same thing in different language. Or, they are saying it happens that way because that’s the way it is. I’m seeking something that will evoke in me something that says that it is reasonable, if you think about it this way.
E.g. - when the two jars reach the bottom and begin to roll, the lighter one rolls farther. This is reasonable when you realize that friction will slow them both down, but the heavier one will create more friction and therefore not roll as far. Is there something that’s REASONABLE that explains why the water filled jar would get rolling a bit faster than the empty one? Or is this really one of those things that stops us right off the bat and says, well, that’s just the way the universe is constituted?
The heavier jar’s greater inertia overcomes air and rolling friction.
First of all, forget about mass. The primary force acting on both jars (gravity) is proportional to mass, so the accelerations (and hence velocities and times to reach the bottom) don’t depend on mass at all (neglecting friction).
But what does matter is how the mass is distributed. With the jar filled with water, most of the mass is relatively near the center of the jar. So most of the mass moves in more or less a straight line down the slope. But with the empty jar, all of the mass is far from the center. So as it rolls down, that mass is moving on a much wilder path. It takes energy to move that mass on a wiggly path, and that’s energy that’s not going into the velocity down the ramp. So even though both jars will have the same energy per mass at any given point on the ramp, with the empty jar, a lot of that mass isn’t being used. So the empty jar goes slower.
Consider a jar full of water (make that frozen water, so we don’t have to worry about fluid and viscous effects) vs a jar made of lead with equal mass. You can even posit a fine glass coat on the lead jar, so surface friction is the same.
Now all the linear forces are the same, and the difference is solely rotational. Which jar would be harder to spin or roll? The lead jar, which has a higher moment of inertia because almost all its mass is farther from the axis of rotation (Think of whirling a weight on a long string, vs a short one – which is harder to spin up to a given RPM?) Conversely, a lead jar rolling at the SAME velocity (same RPM) as a jar of frozen water will keep rolling longer/farther, due to its greater angular momentum
Since yours is a question of intuition, I thought alternate cases might halp you frame/train your intuitions
This, I had. That’s part of the cognitive dissonance this produces.
“wiggly path”??
“…with the empty jar, a lot of that mass isn’t being used.” ??
I’m not quite there. I understand we’re dealing with distribution of mass. I just don’t see how it affects acceleration.
If you look at one spot on the surface of the empty jar as it rolls down the ramp, that spot goes up and down as the jar rotates. Because most of the jar’s mass is on the outside, energy must be used to make that rotation occur. That’s less energy to accelerate the jar down the ramp.
Since the filled jar has most of its mass toward the center, less energy (proportionally) is needed to “lift” it up as it rolls, meaning it can accelerate faster.
Sorry, that would be confusing. Replace that with “…a lot of that energy isn’t being used”.
Ok, to address this aspect particularly consider the simplest case for rotational mechanics: two point masses separated by a massless rod of length L. You can model this at home by using a ballpoint pen with blobs of clay at each end (provided the blobs of clay are much heavier than the pen!).
Now consider that we apply some fixed torque to this object, say by using our fingers near the middle of the pen to start it rotating around its center. Note that at a constant angular velocity, the linear speed of the clay blobs depends directly on their distance from the center of the rotation: if L is large, the masses have a much larger linear velocity than if L is small. Thus, the further the clay blobs are from the center of mass the more kinetic energy they will have for a given angular velocity.
In order to accelerate the system to a given angular velocity, then, more energy is required if L is large than if L is small. This is the effect that you’re observing with the ramp; both jars experience the same torque due to their weights but because the mass of the water-filled jar is closer to its center of mass, the same torque produces a larger acceleration.
Except that they do not experience the same torque. The heavier jar experiences more torque.
The first part of your post does a good job of explaining the concept of angular momentum, the farther something is from the center of rotation, the harder it is to start it spinning or to stop it spinning.
The formula is T=Ialpha* where T is Torque, I is Moment of Inertia, and alpha is Angular Acceleration.
You see that if you increase Torque but do not change the structure of the object (which gives you I), then you increase its angular acceleration. This is intuitive. It also tells you that if you keep the torque the same and increase I (use a jar of the same weight but larger diameter), the object will have a lower acceleration.
In the case of the OP, when you add water to the jar you are affecting both sides of the equation. Both T and I are going up. The point is that when you add mass near the center of rotation of the jar it is easier to start that mass spinning than the original mass of the jar (because the jar material is farther away from the center of rotation than the water), so I only goes up a little compared to your increase in T.
So, big increase in T, little increase in I for:
T=Ialpha*
means that alpha gets bigger, or the jar accelerates more.
In plain words summary, adding material inside the jar increases the torque trying to make it roll more than it affects the jar’s resistance to rolling.
Conversely, adding material to the ouside of the jar should increase its resistance to rolling more than it increases the torque trying make it roll and would cause the jar take more time to get to the bottom.
Another way to look at the problem is to change the way you think about the jars. It’s no longer a “full” jar and an “empty” jar. It’s a weightless jar with extra weight in the middle, and a weightless jar with extra weight on the perimeter. Both jars weigh the same in total. It’s irrelevant, yes, but when you set them equal, it’s even easier.
The perimeter weight takes energy to rotate, the center weight does not. The center weighted jar translates all of its potential energy into forward motion. The perimeter weighted jar translates some of its potential energy into rotational motion, and some into forward motion. Since the total energy is exactly equal at every point (equal mass), the center weighted jar will move faster at every point down the slope.
Back to the real world, your two jars are just different points along the continuum of 100% center weight and 100% perimeter weight.
The way I work it out is:
Acceleration of the empty glass is: ae = mjgIj mg, mj is mass of jar, g is acceleration of gravity, Ij is moment of inertia of the jar.
Acceleration of full jar is: af = (mj + mw)*(Ij + Iw)*g, mw is mass of the water, Iw is moment on inertia of water.
multiplying out the acceleration when full: af = (mjIj + mjIw +mwIj + mwIw)*g
If I now divide af by ae I get. af/ae = 1 + a bunch of positive terms. The ratio is clearly greater than 1 so the acceleration when full is greater than the acceleration when empty.
So it rolls faster.
Thank you. But this is exactly what I want to avoid. This is rational in the sense that it is clearly logical mathematically. What my OP was about is trying to gain some reasonable, intuitive way of thinking about this difference. You have merely converted the difference into mathematical terms. Correct, of course, but you are simply restating mathematically the problem that I’m trying to understand, not explaining it.
And I think I FEEL something in the explanation that talks about the difficulty of imparting acceleration to something “long” with “weights on the end, and no weight in themiddle.” That gets a little closer to what I’m seeking.
You have a meter stick, it’s fixed at one end and spinning 1 time per second. Put a weight 1cm from the center, that weight will be moving very slowly, 6.28cm per second. Put that same weight at the end of the stick, it will be moving 628cm per second. Weight near the axis of rotation moves very slowly around the axis, and has very little rotational energy/momentum. Weight far away from the axis moves quickly and has much more energy/momentum while rotating at the same RPM.
Good. That’s clear. And now, if you will, connect this to the question.
In that case Cheesesteak’s explanation comes close. You have added a lot of torque in addin the wight of the water without adding a proportional inertia since the water’s mass is closer to the center.
Maybe another way to look at it is that the water’s weight acts on the full radious to add torque but acts only on the radius of gyration, which is smaller than the radius, to add inertia.
P.S. The radius of gyration is the distance from the center at which a point mass equal to that of the water would have the same moment of inertia as the water.
The empty jar is like a bunch of weightless sticks with 1Kg of weights on the end. The full jar (actually my center weighted version) is a bunch of sticks with 1Kg of weights in the middle. The sticks are 1 meter long
If they’re both rotating once per second they will both be moving 6.28 meters/sec and have an angular velocity of 6.28 radians/sec
Full Jar - linear kinetic energy 19.7 Joules = 1/2 M*V[sup]2[/sup] = 1/2 *1Kg * (6.28m/s)[sup]2[/sup] ; rotational kinetic energy = 0 J; total energy 19.7 J ;100% linear, 0% rotational
Empty Jar - linear kinetic energy 19.7 J ; rotational kinetic energy = 19.7 J = 1/2 * (moment of inertia[sup]**[/sup]) * (angular velocity)[sup]2[/sup]; total energy 39.4 J ; 50% linear, 50% rotational
For this extreme case, the full jar has 100% of its energy going to forward motion, the empty jar has only 50% going to forward motion, the rest is sucked up by the rotation. Since both jars have the same total energy at each point along the way, the full jar will go faster because more of its energy goes to forward motion.
**for the empty jar, the moment of inertia = length of stick (1m) * mass(1Kg) = 1
for the full jar, the moment of inertia is 0m * 1Kg = 0
Imagine that you have two tops.
You top you spin on your hand. The other one you tie a string to and swing it around in a big circle around your body. If in both cases the top experiences the same number of rotations per a second, then in which case are you using more energy? Well obviously it takes more energy to swing it around your body just to rotate it, as you’re not just rotating the top, you’re also swinging it around.
Now, in the case of the jars, we’re doing the same thing, just in this case we can’t make the two items rotate at the same speed since they both have the same amount of energy (that imparted by gravity.) If they have the same amount of energy, and one needs more energy to rotate, it will go slower.
Apologies - I’m aware of what you say, but it went wrong in my editing. What I had intended to write is that they would experience the same torque per unit mass; in other words, that the actual mass cancels out of the expression for angular acceleration entirely.
I would disagree with your summary here; the increase in torque due to increased mass is irrelevant to the problem, as the increased mass increases the moment of inertia by exactly the same factor it increases the torque. A lead jar with the same mass as the water-filled jar but a different mass distribution would still experience a smaller angular acceleration.
The water-filled jar accelerates down the slope faster purely due to the mass distribution (in the usual language, the lower moment of inertia per unit mass). Those following along at home with mechanics textbooks should note that the situation would be more dramatic if the rotation was centered around the jar’s center of mass - look up the parallel axes theorem for details of how that changes the situation.
I don’t know about other respondents, but I certainly found it a great challenge to explain this all without resorting overly much to technical terminology - apologies for any infelicities that remain as a result!
Cheese, here’s my problem with your (pretty clear) explanation. It’s not an explanation. It simply describes the facts in formula language - a formula that you have not derived yourself, I imagine, by observation. It’s not intuitive to employ the (formerly discovered) formula for kinetic energy here. It’s a learned response and it just states the observation in another language. I’m not discounting the relative simplicity of the two formulas and the ease of comparing them. I’m discounting that it’s natural to understand the situation in formulaic language. I’m guessing that there’s a familiar model or analog for us that will help us understand the phenomenon. Clay on sticks is a good start.