# Effect of density on downward rolling spheres

## Homework Statement

If two balls, being identical in volume, but different in density (one ball is made of iron, the other of aluminum) roll down from an inclined plane, which will reach the bottom first and which will cover a larger distance after having reached the bottom?

IMPORTANT NOTE: please take into account that I need to be able to solve similar question conceptually. I really wonder about the theory, but the practical view on the matter is utmost important as I am not allowed to use a calculator and paper when answering this and similar questions.

## Homework Equations

Moment of inertia (or rotational inertia) (being I = 2/5MR^2 for a solid sphere)
Vector calculations for the forces

## The Attempt at a Solution

My assumptions:
Regarding weight and volume: [/B]I think both weight and volume is important when it comes to the velocity the balls will reach and the distance they will cover.

Regarding air resistance: since the volumes are equal, I assume the air resistance (drag) force acting on it will be equal in size as drag is dependent on area (Fd = ½ ρ * v^2 * Cd * A, with FD = Drag force, ρ = fluid density, v = Relative velocity between the fluid and the object, Cd = Drag coefficient and A = Transversal area or cross sectional area). I even think, that air resistance won’t have any significant effect, because of the low velocity. To reach a high velocity, very long distances would be needed, which seems to me illogical for just an inclined plane.

Regarding the gravity force: The x (or friction) and y component of the gravity is different in size for each ball, but the ratio between them is equal for both. Therefore, the “relative” friction is the same for both (I wonder if the impact of friction in that case is the same for both balls being of the same volume. Anyone who would like to explain?).

Regarding the moment of inertia (or rotational inertia): rotating objects have a moment of inertia (I). The “I” differs per body type. For a solid sphere, I = 2/5MR^2. I think the iron ball with the greater density, having a greater mass, will have a greater “I” compared to the aluminum ball. Torque, as we know, defines acceleration and thus velocity. Since torque can be expressed as Ffriction*Radiusofball = I*angularvelocity, “I” must play a role in defining the acceleration and the velocity at which an object will roll down an inclined plane. Thus, the bigger “I”, the bigger the Torque, the greater the acceleration, the greater the velocity. Therefore, the ball with the greater mass will have a bigger “I”, wherefore it will roll down faster.

For the same reason, a ball with a bigger Volume, having a larger radius, would roll down faster than a ball with an equal mass. Moreover, experiments have proven that a larger ball (identical masses) will roll down faster and further. Is the reason for the greater distance coverage, the greater inertia? Could a greater contact surface between ball and ground also have a significant effect on this?

My question to you: Is it right to conclude it like this or am I underestimating/overseeing determinative factors. I especially care about your conceptual/practical view on the matter. Hoping to receive your valuable guidance. Thanks in advance!

Kind regards

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Therefore, the ball with the greater mass will have a bigger “I”, wherefore it will roll down faster.
I think this line of logic was flawed. You indicate that I is proportional to mass. Where does the torque to rolI the ball come from? Is the torque proportional to mass? If they are both proportional to mass how much does mass affect angular acceleration

Physicsterian
haruspex
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No, you mean angular acceleration.
the bigger “I”, the bigger the Torque
For a given acceleration, a larger I implies a larger torque, but that is telling you it would need a larger torque to accelerate as fast. Your reasoning is as though a larger I causes a larger torque.
What does cause the torque here?

Physicsterian
I think this line of logic was flawed. You indicate that I is proportional to mass. Where does the torque to rolI the ball come from? Is the torque proportional to mass? If they are both proportional to mass how much does mass affect angular acceleration
@haruspex You're right, it had to be "angular acceleration"

The friction Force is responsible for causing the torque. The friction force can be defined as normal Force x static friction constant. And the magnitude of the normal Force gets larger with mass. Therefore, it seems to me that torque must be proportional to mass. And, "mass" is a variable in "I", wherefore I tend to think that a higher "I" should result in more torque.

However - at the other hand:

In a video by Crashcourse it is stated that when the moment of inertia of a body is small, its velocity can take up a larger portion of its kinetic energy, resulting in a higher velocity down the inclined plane. (This proves that the opposite of my assumption should be correct, namely: large "I" -> less torque -> reduction in "velocity)

Lastly, I watched some experiments carried out by prof. W. Lewin, which shows that weight seems to have no effect on the velocity at which cylinders roll down an inclined plane. So, I assume the same must apply to spheres then.

It feels like I am doing something wrong in interpreting the torque equation mathematically...

tnich
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@haruspex You're right, it had to be "angular acceleration"

The friction Force is responsible for causing the torque. The friction force can be defined as normal Force x static friction constant. And the magnitude of the normal Force gets larger with mass. Therefore, it seems to me that torque must be proportional to mass. And, "mass" is a variable in "I", wherefore I tend to think that a higher "I" should result in more torque.

However - at the other hand:

In a video by Crashcourse it is stated that when the moment of inertia of a body is small, its velocity can take up a larger portion of its kinetic energy, resulting in a higher velocity down the inclined plane. (This proves that the opposite of my assumption should be correct, namely: large "I" -> less torque -> reduction in "velocity)

Lastly, I watched some experiments carried out by prof. W. Lewin, which shows that weight seems to have no effect on the velocity at which cylinders roll down an inclined plane. So, I assume the same must apply to spheres then.

It feels like I am doing something wrong in interpreting the torque equation mathematically...
It seems to me that you are only considering one side of the equation. What is the effect of the mass on the other side of the equation?

Delta2
jbriggs444
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The friction force can be defined as normal Force x static friction constant.
That is the limit beyond which the mating surfaces begin to slip. The actual force of static friction is whatever force (within that limit) is required to prevent slipping.

Physicsterian and Merlin3189
haruspex
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mass" is a variable in "I", wherefore I tend to think that a higher "I" should result in more torque.
This is such a standard logical fallacy that it has a Latin name, cum hoc ergo propter hoc.
V=IR, but if I connect a 1.5V battery across twice the resistance it doesn't make it a 3V battery.

Physicsterian and Merlin3189
Delta2
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If you write down the equations of motion (applying the equations ##\sum F=ma## and ##\sum T=I\alpha## for this specific problem) you will find out that the mass gets simplified from both sides of the equations thus it doesn't play a role on the acceleration (and hence on the final velocity). That is because the left side of those equations are appearing quantities that all are proportional to mass, and in the right side (provided that the moment of inertia I is such that is proportional to mass) also we have quantities proportional to mass, so mass gets simplified.
If we don't neglect air resistance then mass doesn't get completely simplified because air resistance force isn't proportional to mass, so it plays a role in this case, but it will be small as long as air resistance is small.

Physicsterian
First of all, thanks everybody for the directions. Hereunder my refined assumptions based on your input:

Regarding the relationship between torque and moment of inertia: @haruspex Applying your analogy to my topic question, it would mean that the torque should be seen as a constant (?) and that, therefore, the angular acceleration must get smaller when the moment of inertia gets larger. Just like the amperage becomes smaller when the resistance increases at a constant voltage supply, in your example

V = IR
50 v = 5 A *10 Ohm
50 v = 2,5 A * 20 Ohm

Also, I tried rearranging the my initial torque formula to see if it could help me in arriving at a logical conclusion: Dividing both sides by “I” left me with “Ffriction*R / I = angular acceleration” or, put differently, “Ffriction/I + R/I = angular acceleration. Am I correct in saying that another way of stating this is that angular acceleration is inversely proportional to “I”? Thus, an increase in “I” would mean a smaller angular acceleration. This is also in some way in accordance with the analogy that @haruspex provided. Am I concluding it correctly so far?

Regarding mass and volume: at the meantime I continued with searching on rotational inertia and found out how the mass (indeed) cancels out and has no effect on the other side of the equation when you substitute angular acceleration with “tangential acceleration / R” and subsequently Ffriction with “μ*Fnormal”. I also saw that the radius also cancels out and thus volume should not have any effect on angular acceleration too. Thanks you for your view @Delta² ; seeing it from a proportionality point of view helps a lot.

On the whole, if all my conclusions above are correct, I see the relationship between rotational inertia and angular acceleration, and also see that mass and volume is of no effect.

Last, but not least, regarding my assumption about the distance that each ball will cover after having reached the bottom: Now, the last thing I wonder is, if I am right in saying that both balls with a different density will also cover the same distance because of the above mentioned conclusions; hat the weight is larger, will also mean that the friction force will be larger, resulting in a the same angular acceleration for both balls and an equal distance being covered.

Delta2
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Mass and Volume have no effect on angular acceleration indeed, however, radius R will have an effect on the final velocity since the tangential acceleration is equal to angular acceleration times R (that is of course if we consider that the motion is such that we have rolling without slipping). So with bigger radius R we ll have bigger final velocity.

As to which object will go further. Friction alone cannot alter the velocity of a sphere that is rolling without slipping (there are many threads on these forums why this is true) (if you ask me, how friction plays a role while the object is moving downwards along the incline while rolling without slipping, the answer is that friction is not the only force there, there is the tangential component of the force of weight). Rolling resistance or of course Air resistance can do that. If we neglect Rolling resistance and Air resistance , then both of the spheres simply will not stop, even if there is friction!

haruspex
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Applying your analogy to my topic question, it would mean that the torque should be seen as a constant (?)
No, the point of the analogy was that if you have a relationship between three variables then you cannot immediately assert a relationship between two of them in a particular situation without considering how the third may vary.
You had effectively assumed angular acceleration was constant to deduce that torque was proportional to I. Instead, you need to consider how the density affects the torque and, independently of that, how density affects I. When you have done that you can use τ=Iα to find how density affects α.

@haruspex

Regarding relationships between 3 variables

I obtained information about relationships among three variables and I realized that I indeed appear to have made a correlation-related misinterpretation/mistake.
So, basically, if you have a fomula like f “Ffriction*R = I * R", you cannot draw conclusions about causality and proportionality straight away. Stating that one entire side of an equation (which represents torque in this case) stays constant, would be incorrect. If you rearrange such that you get only one variable at one side, you can interpret that variable as a constant. To make conclusions about proportionality thereafter, requires, as I understand, knowledge on the physical laws around the specific subject (in this case, rotational motion), and in mathematical terms: knowing the type of relation between predictor, response and covariate. Do I understand it correctly that a/the proper manner to get to see the effect of each variable is to figure out what their individual effect is?

@Delta²

The effect of R on angular acceleration and covered distance

It tried proofing myself what the effect of R is by doing the following: (Please see right side of picture.) First, I substituted the “tangential acceleration” with “angular acceleration*R”. I rearranged the formula such that R got at one side. It seems to me that the fact that R can be put at one side of the eventual formula and that it cannot cancel out, is the evidence that R must have an effect on angular acceleration. Conclusion: it should have an effect. However, also here, it seems to be me that it isn't possible to conclude straight away what the exact relationship between the variables is as there, again, are three variables. Therefore, I get hung at proofing mathematically that a larger radius will cause a larger acceleration and a larger distance coverage.

Something I fail to get is, if volume has no effect, how does radius do? Say the volume increases, then, the radius must also become larger. So, both volume and radius should have an effect, just the fact that volume is not used in the formulas should be what's different then, shouldn't it? Or is there something I fail to see?

And lastly, I have watched an experiment of prof. Lewin in which a disc with a larger radius reaches the bottom at the same time as one with a smaller radius; since the theory behind discs is (as far as I know) the same for spheres, it seems to me that this contradicts the idea that the radius should have an effect ().

@haruspex

Defining the effect of density on angular acceleration:
- Individual effect of density on torque:
It seems to me that torque is directly proportional to density , as Ffriction, which is part of the torque formula, becomes larger with weight (caused by an enlargement in density);
- Individual effect of density on "I": In contrast, torque is inversely proportional to moment of inertia, because an increase in mass in “I = mR2” (caused by an increase in density) will make “I” greater, when the radius does not change.
- Net effect of density on acceleration: Now, assuming “Ffriction * Radius / moment of inertia = angular acceleration”, it seems to me that an increase in density will effect both the "Ffriction * Radius" part and the "moment of inertia part" of the formula equally. Therefore density will cancel out and won’t have any effect on angular acceleration.

Am I doing it right so, or am I at least at the right track?

Thanks a lot.

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Delta2
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OK sorry I was wrong, R doesn't affect tangential acceleration but it affects angular acceleration. But be carefull on whats happening after the rolling on the inclined has ended, friction alone cant make the rolling sphere stop on the horizontal plane.

Last edited:
Physicsterian
haruspex
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Do I understand it correctly that a/the proper manner to get to see the effect of each variable is to figure out what their individual effect is?
You need to identify the independent variables (the inputs) and see how varying exactly one of those affects the dependent variables (the outputs).
However, this is not always straightforward. In the present case, you might consider any two of radius, mass and density as independent, making the third a dependent variable.

Physicsterian
OK sorry I was wrong, R doesn't affect tangential acceleration but it affects angular acceleration. But be carefull on whats happening after the rolling on the inclined has ended, friction alone cant make the rolling sphere stop on the horizontal plane.
I have been being busy with searching on rotational motion today. Since I am pretty new to this subject. I think I need some time to understand all the principles and factors effecting the rolling motion. So, I beg your pardon if my understanding does not go that optimal yet. Anyway, with my current gathered knowledge, it seems to me that the factors that will influence the covered distance after down the inclined plane are the factors that have an influence on linear motion, so:
- Fgx
- Fstaticfriction
- torque produced by angular acceleration
- air resistance
- rolling resistance
However, I get stuck at how to define their eventual effect on distance coverage. I noticed in a many videos that the conservation of energy law is used to calculate for the final velocity. When doing this they look at how potential is being translated into translational and rotational kinetic energy (also there m and R cancel out). So, I know how to calculate the final velocity now. However, I now get stuck at calculating or conceptually estimating the distance coverage. Typing in search terms such as "which ball does will cover a larger distance along an inclined plane" or "rotational motion distance spheres" haven't helped.