About torque and moment of inertia

[Sundown444]

I know I just posted a topic, but I have another question. From my understanding, torque is force times the length of the lever arm, and the longer the lever arm or the more force there is, the more torque is applied. For moment of inertia, it is mass times the square of the distance from the axis of rotation, though it may vary with different shapes. So double the mass, double the moment of inertia, while doubling the distance quadruples the moment of inertia. That said, is moment of inertia usually bigger than the torque in things such as the human body and its limbs? Or is it a different case?

 

[scottdave]

To say one is bigger than the other, you cannot compare moment of inertia to torque as greater or lesser than. They are not the same dimension. It would be sort of asking is force bigger than mass, for example.

 

[Fewmet]

That's really an apples-and-oranges kind of thing: you cannot compare the two values any more than you can compare area to volume.

You can say this: If you have two rings of radius R and 2R and you apply a force F to the edge of each ring, the torque on the larger ring is twice as great but the resulting acceleration of the larger ring is half that of the smaller one.

 

[NFuller]

As others here have said, torque and moment of inertia are two different things. They do relate to each other through the following expression
$$\vec{\tau}=I\vec{\alpha}$$
where $\tau$ is the torque applied to an object with moment of inertia $I$ leading to an angular acceleration $\alpha$. This is effectively the rotational analogue of Newtons second law
$$\mathbf{F}=m\mathbf{a}$$

 

[Sundown444]

I knew that. I meant in terms of the human body. The longer the lever arm, the more torque, but the longer the radius, the more moment of inertia, and thus the more torque is required to cause a change in angular acceleration. I am trying to make sense of something here. When something is longer, it can apply more torque, right? But if the radius of the moment of inertia is longer, it will require more torque. And the radius in moment of inertia is squared. So if something is long, and can apply more torque, will it take even more torque to rotate something due to moment of inertia having its radius squared and all? Sorry if I still am not understanding this well.

 

[Fewmet]

When something is longer, it can apply more torque, right?

You are being imprecise. I know from many years of teaching this stuff that imprecision often hides and exacerbates misunderstanding, so let me clarify.

Are you thinking, for example, that a longer bat can apply a greater torque to another body than a shorter bat when either strike the other body? That might be true or it might not, depending on bat density, rotational speed and time of contact.

Or are you thinking that a longer bat needs a lesser force applied to it to experience a given torque.

I am guessing you mean the latter.

But if the radius of the moment of inertia is longer, it will require more torque.

Does this rewording get at what you mean? If R in the moment of inertia calculation is larger and M remains the same, it will require more torque to produce a given rotational acceleration of the body. That statement is correct.

(Note that is different from having a longer bat becasue the M is probably larger.)

And the radius in moment of inertia is squared. So if something is long, and can apply more torque, will it take even more torque to rotate something due to moment of inertia having its radius squared and all?

It'll probably help to think in terms of a specific experiment, and you asked about the human body. One complication that introduces is that the typical force responsible for raising an arm is applied by muscles near the shoulder, not at the fingertip. Making an artificial situation that perhaps meets your criteria, suppose a man is standing with his arm hanging vertically downward. You apply a force F to his fingertips to raise the arm 90˚ in in time ∆t.

What happens if the man now has his hand on his shoulder (so the folded arm is vertical and the elbow points down) and you apply F to his elbow? The mass of the folded arm is the same as the extended arm. The force F will generate roughly half the torque that it did before. The shorter arm has roughly a quarter of the moment of inertia. The folded arm is lifted 90˚ in less time than the extended arm.

 

[Sundown444]

You are being imprecise. I know from many years of teaching this stuff that imprecision often hides and exacerbates misunderstanding, so let me clarify.

Are you thinking, for example, that a longer bat can apply a greater torque to another body than a shorter bat when either strike the other body? That might be true or it might not, depending on bat density, rotational speed and time of contact.

Or are you thinking that a longer bat needs a lesser force applied to it to experience a given torque.

I am guessing you mean the latter.Does this rewording get at what you mean? If R in the moment of inertia calculation is larger and M remains the same, it will require more torque to produce a given rotational acceleration of the body. That statement is correct.

(Note that is different from having a longer bat becasue the M is probably larger.)It'll probably help to think in terms of a specific experiment, and you asked about the human body. One complication that introduces is that the typical force responsible for raising an arm is applied by muscles near the shoulder, not at the fingertip. Making an artificial situation that perhaps meets your criteria, suppose a man is standing with his arm hanging vertically downward. You apply a force F to his fingertips to raise the arm 90˚ in in time ∆t.

What happens if the man now has his hand on his shoulder (so the folded arm is vertical and the elbow points down) and you apply F to his elbow? The mass of the folded arm is the same as the extended arm. The force F will generate roughly half the torque that it did before. The shorter arm has roughly a quarter of the moment of inertia. The folded arm is lifted 90˚ in less time than the extended arm.

For the last part, can you please try to explain it in simpler terms? I think I understand, don't get me wrong, but I just want to make sure.

 

[Fewmet]

Can you say specifically which part you'd like clarified? The arm positions in the experiment? Why it takes less time to lift the folded arm?

Would showing the equations help? (Some people would find that simpler and some more complicated...)

 

[Sundown444]

Both the arm posit

Can you say specifically which part you'd like clarified? The arm positions in the experiment? Why it takes less time to lift the folded arm?

Would showing the equations help? (Some people would find that simpler and some more complicated...)

Both the arm positions and why it takes less time to lift the folded arm. And why the force will generate half as much torque as it did before.

Show me the equations, too.

 

[Fewmet]

The first trial is with the man standing and letting his arms hang straight down, so his fingertips point at the ground (like this). You apply the force F tangentially to the fingertips on, say, his right hand and raise it in a smooth arc until it is horizontal, so you have raised it 90˚.

I cannot find an image online for the arm position in the second trial, but think of touching your right shoulder with your right hand while keeping the elbow pointing down. That pretty much halves the length of the arm but keeps the mass the same. Now the force is applied tangentially to the elbow to raise it until horizontal.

As I think you understand, in the first trial the lever arm is greater and (since torque is lever arm times perpendicular force), the torque applied to the arm is greater than in the second trial. Focus on toque as something that causes a rotational acceleration (just as a translational force acting on the center of mass of a body makes it accelerate translationally).

As you also see, the lever arm of the folded arm is shorter, so the moment of inertia is smaller. (Treating the arm as a cylinder rotating about its end
I=MR²/3.) Halving the R (so substituting R/2 and then squaring). the moment of inertia is one quarter as great.

Focus on moment of inertia as a resistance to a change in rotation, or as a resistance to the effects of a torque. Halving the arm length increases the torque acting on the arm and decreases the resistance of the arm to the torque's effects, so the rotational acceleration is greater.

Finally, both arms start at rest and swing through the same angle. The shorter arm "speeds up" more rapidly. so it reaches the horizontal position more quickly.

Does that help?

 

[Sundown444]

The first trial is with the man standing and letting his arms hang straight down, so his fingertips point at the ground (like this). You apply the force F tangentially to the fingertips on, say, his right hand and raise it in a smooth arc until it is horizontal, so you have raised it 90˚.

I cannot find an image online for the arm position in the second trial, but think of touching your right shoulder with your right hand while keeping the elbow pointing down. That pretty much halves the length of the arm but keeps the mass the same. Now the force is applied tangentially to the elbow to raise it until horizontal.

As I think you understand, in the first trial the lever arm is greater and (since torque is lever arm times perpendicular force), the torque applied to the arm is greater than in the second trial. Focus on toque as something that causes a rotational acceleration (just as a translational force acting on the center of mass of a body makes it accelerate translationally).

As you also see, the lever arm of the folded arm is shorter, so the moment of inertia is smaller. (Treating the arm as a cylinder rotating about its end
I=MR²/3.) Halving the R (so substituting R/2 and then squaring). the moment of inertia is one quarter as great.

Focus on moment of inertia as a resistance to a change in rotation, or as a resistance to the effects of a torque. Halving the arm length increases the torque acting on the arm and decreases the resistance of the arm to the torque's effects, so the rotational acceleration is greater.

Finally, both arms start at rest and swing through the same angle. The shorter arm "speeds up" more rapidly. so it reaches the horizontal position more quickly.

Does that help?

I understand. Except for one thing: The part where halving the arm length increases the torque. Why exactly is that so? I thought a longer lever arm meant more torque, or are we considering something else?

 

[Fewmet]

I understand. Except for one thing: The part where halving the arm length increases the torque. Why exactly is that so? I thought a longer lever arm meant more torque, or are we considering something else?

Oops: my mistake. I am trying to do too many things at once at the moment. You are right: pushing at the elbow produces a smaller torque. If we treat the lever arm as being cut in half, the torque is cut in half and the moment of inertia is quartered (for the reasons that motivated your original question). Revising my sentence from my previous post,
Halving the arm length halves the torque acting on the arm and quarters the resistance of the arm to the torque's effects, so the rotational acceleration is greater.

Thanks so much for catching that.

 

[Sundown444]

Oops: my mistake. I am trying to do too many things at once at the moment. You are right: pushing at the elbow produces a smaller torque. If we treat the lever arm as being cut in half, the torque is cut in half and the moment of inertia is quartered (for the reasons that motivated your original question). Revising my sentence from my previous post,
Halving the arm length halves the torque acting on the arm and quarters the resistance of the arm to the torque's effects, so the rotational acceleration is greater.

Thanks so much for catching that.

You're welcome.

And lengthening the arm doubles the torque and (quadruples?) the moment of inertia as well, if I have this down correctly?

 

[Fewmet]

Correct, but I again emphasize the point of precision. Doubling the arm length doubles the torque you apply to the arm for a given force and quadruples the moment of inertia of the arm.

As an aside, a cool application of all this is seen in walking and running. A simple model of walking is that the leg is a cylinder allowed to swing forward under a torque applied by gravity. A longer leg experiences a smaller rotational acceleration and takes longer to swing through a given angle (i.e., a step takes longer). As a tall person, I am keenly aware of this when I walk beside someone much shorter. You also see in in the pace of a Great Dane versus a Pekingese.

When you run, you want the time of swing to be shorter. You accomplish that by shortening the lever arm (bending your leg so the part below the knee swings faster).

 

[Sundown444]

Correct, but I again emphasize the point of precision. Doubling the arm length doubles the torque you apply to the arm for a given force and quadruples the moment of inertia of the arm.

As an aside, a cool application of all this is seen in walking and running. A simple model of walking is that the leg is a cylinder allowed to swing forward under a torque applied by gravity. A longer leg experiences a smaller rotational acceleration and takes longer to swing through a given angle (i.e., a step takes longer). As a tall person, I am keenly aware of this when I walk beside someone much shorter. You also see in in the pace of a Great Dane versus a Pekingese.

When you run, you want the time of swing to be shorter. You accomplish that by shortening the lever arm (bending your leg so the part below the knee swings faster).

So, given a force for torque and mass for moment of inertia along with the radius for both, depending on the force and mass for their respective equations for calculating torque and moment of inertia, moment of inertia can be bigger than torque when you lengthen the arm but torque is bigger than moment of inertia when you shorten the arm, is that right?

 

[Fewmet]

So, given a force for torque and mass for moment of inertia along with the radius for both, depending on the force and mass for their respective equations for calculating torque and moment of inertia, moment of inertia can be bigger than torque when you lengthen the arm but torque is bigger than moment of inertia when you shorten the arm, is that right?

There are two problems there. One is that you try to compare two different quantities. Consider that 1.000 cm³ of lead has a mass of 11.34 g. Is it meaningful to say lead has a bigger mass than volume? You can say the numerical value of the mass is greater than the numerical value of the volume, but that will be true or false depending on the units you select. It does not seem to be a very useful statement.

The other problem is that "but" in you last sentence. It implies the relationship changes depending on whether you increase or decrease lever arm. Torque is linearly proportional to lever arm and moment of inertia is proportional to the square of the lever arm whether you increase or decrease lever arm.

Do you have a specific goal in framing the summary in terms of "bigger" and "smaller"? It has the same problem as saying two apples are bigger than two oranges. That might be true of mass or volume but not of citric acid content, flavor, sweetness...comparisons between dissimilar units tend to be pretty ambiguous and not very useful. If you have a specific applicaiton for the summary statement in mind, I may be able to help frame it.

 

[Sundown444]

There are two problems there. One is that you try to compare two different quantities. Consider that 1.000 cm³ of lead has a mass of 11.34 g. Is it meaningful to say lead has a bigger mass than volume? You can say the numerical value of the mass is greater than the numerical value of the volume, but that will be true or false depending on the units you select. It does not seem to be a very useful statement.

The other problem is that "but" in you last sentence. It implies the relationship changes depending on whether you increase or decrease lever arm. Torque is linearly proportional to lever arm and moment of inertia is proportional to the square of the lever arm whether you increase or decrease lever arm.

Do you have a specific goal in framing the summary in terms of "bigger" and "smaller"? It has the same problem as saying two apples are bigger than two oranges. That might be true of mass or volume but not of citric acid content, flavor, sweetness...comparisons between dissimilar units tend to be pretty ambiguous and not very useful. If you have a specific applicaiton for the summary statement in mind, I may be able to help frame it.

I don't think I was directly comparing. I could be wrong, but I guess what I am trying to say is that moment of inertia increases more than torque does when the arm is lengthened?

Also, please tell me, why does moment of inertia decrease by a quarter when the arm is shortened? Please include equations to help describe this.

 

[Fewmet]

I guess what I am trying to say is that moment of inertia increases more than torque does when the arm is lengthened?

I think most people with a technical background would say it in terms of one increasing linearly with lever arm and the other increasing with the square of the lever arm.

Also, please tell me, why does moment of inertia decrease by a quarter when the arm is shortened? Please include equations to help describe this.

For a ring of radius R and mass M rotating about an axis through its center,
I=MR².
If you double the lever arm:
I=M(2R)²=4R², so doubling R quadruples I.
If you cut the original radius in half:
I=M(R/2)²=MR²/4, so halving R quarters I.

Does that make sense?

 

[Sundown444]

I think most people with a technical background would say it in terms of one increasing linearly with lever arm and the other increasing with the square of the lever arm.
For a ring of radius R and mass M rotating about an axis through its center,
I=MR².
If you double the lever arm:
I=M(2R)²=4R², so doubling R quadruples I.
If you cut the original radius in half:
I=M(R/2)²=MR²/4, so halving R quarters I.

Does that make sense?

It does make sense, just two questions, does this apply to other objects as well, since different objects have different moment of inertias? Also, where is the mass M in 4R² equation you put in the second equation you showed?

As for the first part of what you said, about torque increasing linearly with the lever arm and moment of inertia increasing with the square of the lever arm, that was basically what I was trying to ask all along.

 

[Fewmet]

...does this apply to other objects as well, since different objects have different moment of inertias?

Yes. Note, though, that we are talking about uniform bodies in two dimensions experiencing a single torque. For more complicated bodies in more dimensions, you have to allow for variations in shape and mass density (usually requiring integral calculus). For multiple torques, there are techniques for combining them. The principles, however, are the same. As you can see in the application I offered for walking and running, the principle can let you work qualitatively with complicated situations.

As you indicated earlier, different mass distributions have different formulas for moment of inertia. That, of course, has to be taken into account

Also, where is the mass M in 4R² equation you put in the second equation you showed?

I am so bad at proofing my own writing...I mistakenly omitted the M.

As for the first part of what you said, about torque increasing linearly with the lever arm and moment of inertia increasing with the square of the lever arm, that was basically what I was trying to ask all along.

I understand. Again, physicists (and people in technical fields in general) place huge value on precise language. In everyday life we get lax with language but usually have such a deep understanding of a common context that we can figure out the intended meaning. Dealing with an area where there is a huge difference, for example, between "move" and "accelerate" but arguably no difference between "rest" and uniform motion", precise use of technical language can be essential.

 

[Sundown444]

Yes. Note, though, that we are talking about uniform bodies in two dimensions experiencing a single torque. For more complicated bodies in more dimensions, you have to allow for variations in shape and mass density (usually requiring integral calculus). For multiple torques, there are techniques for combining them. The principles, however, are the same. As you can see in the application I offered for walking and running, the principle can let you work qualitatively with complicated situations.

As you indicated earlier, different mass distributions have different formulas for moment of inertia. That, of course, has to be taken into account
I am so bad at proofing my own writing...I mistakenly omitted the M.
I understand. Again, physicists (and people in technical fields in general) place huge value on precise language. In everyday life we get lax with language but usually have such a deep understanding of a common context that we can figure out the intended meaning. Dealing with an area where there is a huge difference, for example, between "move" and "accelerate" but arguably no difference between "rest" and uniform motion", precise use of technical language can be essential.

I understand. So for the second formula you provided, is it M4R²?

 

[Fewmet]

I understand. So for the second formula you provided, is it M4R²?

Yes, it should have read

If you double the lever arm:
I=M(2R)2=4MR², so doubling R quadruples I.

 

[Sundown444]

Yes, it should have read

Okay then.

So why does moment of inertia decrease by a quarter (or a factor of four) mathematically when radius is halved?

 

[jbriggs444]

A fixed torque will produce twice as much tangential acceleration if you cut the radius in half.
A fixed tangential acceleration will correspond to twice as much angular acceleration if you cut the radius in half.

That's two factors of two. Or one factor of four.

 

[Fewmet]

So why does moment of inertia decrease by a quarter (or a factor of four) mathematically when radius is halved?

What jbriggs444 posted is correct. If that is not convincing to you, consider this: You seem fine with doubling the lever arm meaning that the moment of inertia quadruples. It quadruples because of the MR² in the equation tells you multiply by mass by the lever arm by the lever arm. For an arm length of 2R, that means I=M*2R*2R=4MR².

For an arm length of ½R it tell you I = M*½R*½R=¼MR².

Maybe try it with numbers. For a ring with M=4 kg and R=1 m rotating about a axis through it center,
I=MR²=(4 kg)(1 m)²= 4kg-m².
For a ring with M=4 kg and R=½ m,
I=MR²=(4 kg)(½ m)²= 1kg-m².

 

[Sundown444]

What jbriggs444 posted is correct. If that is not convincing to you, consider this: You seem fine with doubling the lever arm meaning that the moment of inertia quadruples. It quadruples because of the MR² in the equation tells you multiply by mass by the lever arm by the lever arm. For an arm length of 2R, that means I=M*2R*2R=4MR².

For an arm length of ½R it tell you I = M*½R*½R=¼MR².

Maybe try it with numbers. For a ring with M=4 kg and R=1 m rotating about a axis through it center,
I=MR²=(4 kg)(1 m)²= 4kg-m².
For a ring with M=4 kg and R=½ m,
I=MR²=(4 kg)(½ m)²= 1kg-m².

So, the moment of inertia increases and decreases by four, and doesn't increase when you both increase or decrease the radius?

 

[Fewmet]

So, the moment of inertia increases and decreases by four, and doesn't increase when you both increase or decrease the radius?

I am not sure of your meaning. Is this consistent with your understanding?
Increasing R increases I. Decreasing R decreases I.
More specifically, Doubling R quadruples I. Halving R quarters I.

 

[Sundown444]

I am not sure of your meaning. Is this consistent with your understanding?
Increasing R increases I. Decreasing R decreases I.
More specifically, Doubling R quadruples I. Halving R quarters I.

Never mind. I did the math and I get it now. Thanks.

 

[ggl205]

Interesting conversation. I was hoping it would eventually get around to cycling and the ongoing debate about different length crank arms.

Historically, sprinters favor shorter crank arms while pursuit and time trial racers like longer crank arms. The physics apply but one variable I find missing or should I say underestimated, is how the longer crank arm affect human energy to move that longer arm through a single spindle rotation. Leaving acceleration aside, increased torque of a longer crank arm will require more energy per revolution than a smaller one. So, if cadence remains the same for longer and shorter arms, energy required to move the longer arm should be greater. If this is true, the commonly held notion that longer crank arms are better for time trial racers may not hold water.

 

[jbriggs444]

how the longer crank arm affect human energy to move that longer arm through a single spindle rotation.

Biomechanics is messy. Details matter.

Are you holding pedal force constant as you increase the crank arm length and maintain cadence? Is that realistic?

 

[russ_watters]

Historically, sprinters favor shorter crank arms while pursuit and time trial racers like longer crank arms. The physics apply but one variable I find missing or should I say underestimated, is how the longer crank arm affect human energy to move that longer arm through a single spindle rotation. Leaving acceleration aside, increased torque of a longer crank arm will require more energy per revolution than a smaller one. So, if cadence remains the same for longer and shorter arms, energy required to move the longer arm should be greater. If this is true, the commonly held notion that longer crank arms are better for time trial racers may not hold water.

No, the power required to move the bike is the same either way. A longer crank arm means a smaller force to achieve the same torque. A longer crank arm would allow you to provide more torque if needed for acceleration.