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Rolling different masses down a hill  Intertia Question 
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#1
Jun409, 07:54 PM

P: 12

I have a very simple conceptual physics problem that I am looking to solve, and unfortunately I've been out of school a little too long to solve:
Suppose you have a hill, and at the bottom of the hill it levels off and keeps going on. You take two marbles of different masses and roll them down the hill with the same force. Which marble will travel the farther distance? Ignore air resistance, AND assume that friction acts on them the same way (i.e., μ is adjusted so that F_f, the force of friction, is the same). My understanding of inertia is that, although both marbles will reach the bottom of the hill with the same speed, the more massive marble will have more momentum/inertia, and will therefore take longer to stop. My sister's 4th grade teacher tried arguing the opposite  that the smaller marble would travel further  but conceptually this doesn't make any sense to me. Second question: if you make the situation more realistic, by keeping μ constant, so F_f is greater for the more massive marble, how do you balance the greater inertia of the larger marble with it's high F_f? That is, will a more massive marble always travel further, or is there some mass ratio between the two marbles where the smaller marble will travel further? Thanks, Jacob 


#2
Jun409, 08:13 PM

P: 505

Actually, if you completely ignore friction and air resistance, the marble doesn't even need to roll!! The marble would simply glide down the slope without rolling. It's the frictional force that makes the marble roll.



#3
Jun409, 08:57 PM

P: 12

I understand the marble won't roll without friction. The point is not to ignore friction entirely  then the question is meaningless  but in the first case, to assume that the friction for the two marbles is constant, and for the second case, to closer simulate reality by allowing the friction to have different values.
In other words, I'm trying to separate the inertial properties from the friction properties to better understand how inertia works. 


#4
Jun409, 09:25 PM

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P: 41,351

Rolling different masses down a hill  Intertia Question
Welcome to the PF, zw. In your thought experiment, what makes the marbles stop after they have reached the flat? There is no air resistance, and apparently no rolling resistance, so they will continue on forever. Oh, and at different speeds, BTW. Quiz Question  why are they moving at different speeds at the bottom of the ramp?



#5
Jun409, 09:31 PM

P: 12

The force of friction slows the balls down. In the first experiment, both marbles have the same friction force working against the marbles (F_f marble 1 = F_f marble 2). In the second case, the more massive marble has a great friction force, because it it more massive, so the normal force is greater.
I am not trying to _ignore_ friction  I am trying to isolate the principal of interia. Re: Quiz question  why would they be traveling at different speeds? If the only force is gravity (assuming F_grav >> F_friction) acting on the marbles down the hill, won't they reach the bottom of the hill, at the same time, with the same speed? Thanks for your questions and the help. 


#6
Jun409, 09:52 PM

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P: 41,351

Friction on the ramp helps spin the balls up. The more massive ball has a higher moment of inertia, so more energy goes into the rotational energy of the heavier ball. The heavier ball also has more potential energy at the top of the release, since the potential energy is mgH, and depends on the mass. Actually, I should spend a bit more time checking the equations  the point I was trying to make with my QQ was that there is more rotational energy in the heavier ball, so there would be less kinetic energy (less velocity). But since the heavier ball has a higher potential energy (PE) initially by virtue of its higher mass, maybe things cancel out. I need to look up the moment of inertia of a sphere. I'll try to do that tonight  but if somebody else beats me to it and can help this poster out with the TE = KE + PE equations, that would be great. 


#7
Jun409, 10:32 PM

P: 4,663

So if the hill height is h, then mgh = (2/10) mv^{2} + (1/2) mv^{2} = (7/10)mv^{2} for both balls. 


#8
Jun409, 10:40 PM

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P: 41,351

Thanks Bob! Okay so to the OP, they will have the same velocity at the start of the flat. But the heavier ball will overcome air resistance and rolling resistance better. But your thought experiment has neither in it...



#9
Jun409, 11:06 PM

P: 34

Neither you nor the teacher are seeing the whole picture: inertia both keeps the ball from rolling when gravity first starts pulling on the heavier object it and keeps it rolling when it reaches the bottom of the hill. Both marbles travel the same distance.



#10
Jun409, 11:08 PM

P: 12

I apologize for the nontechnical terms, for this question assume "resistance" = rolling resistance = rolling friction = rolling drag = whatever force you want to call it that actually slows spherical objects down as they roll.
This is the question: Go to some actual hill. Place two spherical objects of the same size, but of different masses at the top of the hill. Push the marbles down the hill with the same force. Which marble, in real life, will actually go further? Ignore air resistance. How would this sum be quantified? At the crux of the question is this: will the inertia of the larger ball cause it go further, once the balls are done accelerating at the bottom of the hill? Thanks for all the help so far guys. 


#11
Jun409, 11:09 PM

P: 12




#12
Jun509, 02:34 PM

P: 4,663




#13
Jun509, 02:44 PM

P: 12

Thanks for all the help thus far, but I don't believe anyone has answered the question yet.
I am not trying to determine if the balls will slide vs. roll. I'm trying to understand the principal of interia While slide vs. roll, coefficient of friction, friction vs. drag, etc. all might be helpful in other cases, I am trying to keep all other variables the same and ONLY change mass. How does this affect how far the marbles will roll? Go to a hill. Take two marbles of different masses. Roll them down the hill. Which marble will go further? As far as I see it, only one of three things will happen:  The heavier marble roll farther, because it has more inertia at the bottom of the hill;  Both marbles roll the same distance, because this case is independent of mass;  The lighter marble roll farther, because it has less inertia, so it will be carried down the hill faster, and will have more velocity to propel it further; Which one of these three things will happen? 


#14
Jun509, 03:37 PM

P: 4,663

The first is correct. The quantitative definition of "inertia" of each ball is its mass, which is proportional to radius cubed, as is its kinetic energy at the bottom of the hill. The air drag force may at first be the velocitysquared type (like automomiles), which is proportional to radiussquared times velocitysquared, but eventually becomes Stokes' Law drag force, which is proportional to velocity times radius (no squares). So even on a very smooth level surface (but requires rolling w/o sliding), the bigger ball will roll further.



#15
Jun509, 03:55 PM

P: 759

Both balls will roll the same distance. This is the same thing as the inclined plane experiments done by Galileo. You can prove this by having the balls roll back up another inclined plane once they reach the bottom. No matter how steep or shallow the second inclined plane, they will always return to the same height at which they were released (minus losses due to friction). Which means they will both travel the same distance regardless of their difference in mass.



#16
Jun509, 04:10 PM

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P: 41,351

zw  do you understand that there has to be some kind of retarding force in order for the balls to stop? 


#17
Jun509, 04:22 PM

P: 759

I think the OP wanted the question answered both ways. If resistance to motion due to friction is the same for both objects then they will both travel the same distance. In real world experiment there would probably be greater friction for the more massive object.



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