What is the formula for calculating the moment of inertia of a double cone?

[Buzzdiamond1]

What is the correct formula for a double cone, is it 3/10 mr^2, or 3/5 mr^2..?

 

[berkeman]

What is the correct formula for a double cone, is it 3/10 mr^2, or 3/5 mr^2..?

Welcome to the PF.

Can you post your sources for those expressions, or show your work with the integral?

 

[Buzzdiamond1]

Thanks for chiming in Berkeman, much appreciated. I'm not exactly sure what you're asking me, but my question is, what the correct formula for a double cone, is it: I = 3/10 x m x rr or I = 3/5 x m x rr..?

 

[berkeman]

Thanks for chiming in Berkeman, much appreciated. I'm not exactly sure what you're asking me, but my question is, what the correct formula for a double cone, is it: I = 3/10 x m x rr or I = 3/5 x m x rr..?

I understand what you are asking, but I'm trying to get you to help us help you to figure it out. The answer should be available via a Google search, and you offer two possible expressions which means you probably have done those searches. If you could post links to the two different results, we can probably help you to pick the right one.

Otherwise, we can always just do the integration, but I'm lazy and would prefer that you do the integration yourself and post your work so I can check it.

 

[Buzzdiamond1]

I believe the correct formula is 3/5 x m x rr. It would be similar to a sphere and hemisphere, whereas the hemisphere (1/2) is derived from the whole, with the sphere being a standard shape in physics. On the other hand, in the case of the double cone, the single cone is the standard shape, therefore, the formula for the cone has to double for the double cone, correct..?

 

[Buzzdiamond1]

Would you be kind enough to give me your thoughts on this physics report, concerning the double cone..? http://www.lane1bowling.com/pdf/Theoretical_Calculations.pdf

 

[Buzzdiamond1]

Hello everyone, I'm still trying to get a concrete answer to this puzzle. Is the MOI of a double cone 3/10 MR^2 or 3/5 MR^2..?

 

[jbriggs444]

Hello everyone, I'm still trying to get a concrete answer to this puzzle. Is the MOI of a double cone 3/10 MR^2 or 3/5 MR^2..?

So you are talking about a double cone (e.g. a pair of ice cream cones tip to tip) rotating around the central axis of symmetry.

From your searches, you have found a formula for the moment of inertia of a single cone: 3/10 MR².

If you have two identical cones, the moment of inertia should be twice as high as if you had only one, right?
If you have two identical cones, the total mass will also be twice as much as if you had only one, right?

Given this, there is no need to put an added factor of two into the formula. The factor of two is already present in the increased mass.

 

[Buzzdiamond1]

Jbriggs, I appreciate your reply, but let's say the formula for the double cone was 3/5 MR^2. The single cone would be half the total mass, so the formula would also work, correct..? What I'm saying, is that the formula for the double cone is based starting from a single cone or 1/2 of that. Whereas the formula for a full circle is based on a full circle, not from the starting base of a semi circle. Because the single cone is a standard shape calculations are being concluded from that. Conversely, the shpere's MOI wasn't derived from a hemisphere, rather the hemisphere MOI was derived from the whole. Are you understanding my logic..?

 

[Buzzdiamond1]

Here's another thought, let's call it the marble and the top theory. Wouldn't you agree that a spinning top is harder to move than a spinning sphere..? If you agree on that, then the MOI for a top must be higher than a sphere, correct..? Therefore, the MOI for a double cone ( basically a top) has to be higher than the MOI for a shpere. So, if a sphere's MOI is 2/5 MR^2, the double cone and/or a top has to be higher than that, which is why the correct formula should be 3/5 MR^2, not 3/10 MR^2. Your thoughts..?

 

[Nidum]

Edit :

I see from that linked document that you are talking about the base to base configuration .

Confusion arose because 'double cone' has a conventional meaning in geometry and your usage of the words describes something different .

Anyway - don't guess - do the sums .

 

[jbriggs444]

Wouldn't you agree that a spinning top is harder to move than a spinning sphere..?

Why would I agree to that? Per unit mass, the moments of inertia are 0.3 for the cone and 0.4 for the sphere.

 

[Nugatory]

There's a bit of talking past one another going on here, because the answer is different for different axes of rotation. Just to be sure that we're talking about the same problem.

Are the cones joined tip to tip or base to base? Is the axis of rotation a line that passes through both tips and the center of both bases, or is it perpendicular to that line? That's two questions with two answers for a total of four possibilities, and only two of the four have the same answer.

 

[Buzzdiamond1]

Why would I agree to that? Per unit mass, the moments of inertia are 0.3 for the cone and 0.4 for the sphere.

That's if you believe the numbers we've been presented with. I'm contesting those numbers, with my arguments to back it up, along with a physics study report that was performed a while back in my post above dated Nov. 29th. Isn't it common sense knowledge that if you spin a marble and spin a top next to each other, if you blow at them, the marble will be blown off the table and the top will still be standing relatively close to the same position..? Or, if you blow the marble towards the top, upon impact, the marble will go flying, not the other way around..? Again, I don't believe the 3/10 MR^2 formula is correct, hence this discussion.

 

[Buzzdiamond1]

There's a bit of talking past one another going on here, because the answer is different for different axes of rotation. Just to be sure that we're talking about the same problem.

Are the cones joined tip to tip or base to base? Is the axis of rotation a line that passes through both tips and the center of both bases, or is it perpendicular to that line? That's two questions with two answers for a total of four possibilities, and only two of the four have the same answer.

The cones I'm referring to are joined base to base, with the axis of rotation spinning like a top.

 

[jbriggs444]

That's if you believe the numbers we've been presented with. I'm contesting those numbers, with my arguments to back it up, along with a physics study report that was performed a while back in my post above dated Nov. 29th. Isn't it common sense knowledge that if you spin a marble and spin a top next to each other, if you blow at them, the marble will be blown off the table and the top will still be standing relatively close to the same position..? Or, if you blow the marble towards the top, upon impact, the marble will go flying, not the other way around..? Again, I don't believe the 3/10 MR^2 formula is correct, hence this discussion.

The experiments you point to do not test moment of inertia.

Further, any personal speculation you have that 3/10 MR^2 is incorrect is out of place on these forums.

 

[Buzzdiamond1]

With all due respect, the resistance to angular acceleration, angular motion and/or a change in direction, is exactly what my experiments are pointing to. I'm having a tough time understanding why you'e demanding an end this discussion, when clearly my statements have validity..?

 

[Buzzdiamond1]

Is it not possible for a mistake..? Are you suggesting that nothing can change, the Earth is still flat or Einstein wasn't wrong on occasion..? Clearly, these forums do suggest Einsteins formulas are being put to the test and I'm quite sure, he has been wrong at least once in his life, correct..?

 

[Nugatory]

if you spin a marble and spin a top next to each other, if you blow at them, the marble will be blown off the table and the top will still be standing relatively close to the same position..? Or, if you blow the marble towards the top, upon impact, the marble will go flying, not the other way around..? Again, I don't believe the 3/10 MR^2 formula is correct, hence this discussion.

That's not a test of the moment of inertia, it's a test of the surface area, surface velocity, and drag coefficient of the solid (with some more complicated second-order effects thrown in).

Is it not possible for a mistake..?

It is possible, and a good way to check that possibility would be for you to evaluate the integral (as @berkeman suggested in post #4 above) and if you don't come up with the currently accepted results post your work. Either you've found a mistake that has gone undetected for more than three centuries or you've made a mistake yourself - and with the calculation posted people will be able to figure out which it is.

 

[berkeman]

Is it not possible for a mistake..? Are you suggesting that nothing can change, the Earth is still flat or Einstein wasn't wrong on occasion..? Clearly, these forums do suggest Einsteins formulas are being put to the test and I'm quite sure, he has been wrong at least once in his life, correct..?

It is possible, and a good way to check that possibility would be for you to evaluate the integral (as @berkeman suggested in post #4 above) and if you don't come up with the currently accepted results post your work. Either you've found a mistake that has gone undetected for more than three centuries or you've made a mistake yourself - and with the calculation posted people will be able to figure out which it is.

So one last chance -- post your work on the integrals showing the error, or this thread will be closed. We don't let newbie posters come here and show no work and waste the valuable time of our helpers. Please do this...

 

[Buzzdiamond1]

There's been a physic's study outlying the calculations derived manually in his experiment/testing/research from Dr. Joseph Howard at Salsbury University, concluding that the MOI formula for a double cone is 3/5 MR^2, outlined here, http://lane1bowling.com/techdata/core-report/index.html then here http://lane1bowling.com/pdf/Theoretical_Calculations.pdf.

I'm sorry I'm not a physics major, which is why I've come in here to discuss the situation and hope you can appreciate different techniques, as simple as they may be, in finding answers to problems..? As the ole saying goes, there's more than one way to skin a cat. One of the reasons I believe this formula to be correct, is when I first applied for my patent, my patent attorneys son was a physics major at Syracuse University. He also did some calculations, concluding at the time the same formula, which was done 20 years earlier than Dr. Joe's report. So now we have 2 independent studies done 20 years apart, concluding the same results, that I = 3/5MR^2.

The second reason is with me applying some rudimentary common knowledge to the situation at hand. Whereas people know there's a good amount of centrifugal and/or gyroscopic force going on with a top, that a round ball does not possesses when spun. The top doesn't move so easily when you blow on it, compared to a spinning round ball/marble. I would not conclude this is due to more surface area touching the table from the point of the top, compared to the spot on the ball that touches the table, as the surface contact from both objects would be similar, especially if the bottom of the top is rounded like a sphere.

Therefore, my knowledge of basic physics learned very early in life, leads me to conclude that the top is harder to move and/or has more resistance to angular motion, due to having more gyroscopic force and/or a higher moment of inertia.

I also believe the double cone shape hasn't been around for 300 years, because if it was, it would have been considered a standard shape, with a published MOI, which it does not have published. All we have right now are people deriving the double cone, out of the same principles that apply to a single cone, similar to that of a hemisphere and a full sphere, which has shown to be different in Dr. Joe's report.

As stated, how can a round ball/sphere have a higher moment of inertia than a top..? This really makes no sense, other than if ones conclusion is derived from wrong information currently being disseminated..? This is my contention, with the work of Dr. Joseph Howard to back it up.

I've provided a study/report to back up my contention, are you able to provide me a link to another study that shows different calculations..? I'm only trying to expose light on the situation and very much appreciate everyone's time.

 

[jbriggs444]

double cone is 3/5 MR^2, outlined here

If the M is the mass of the single cone, that result would be correct. If the M is the mass of the double cone then it is wrong.

It is fairly obvious that the moments of inertia of an object and of an otherwise identical object which is reflected about a plane at right angles to the axis of rotation will be equal to one another. [Take a cone and flip point to base -- the moment of inertia does not change]

It is fairly obvious that the moment of inertia of two identical objects, both centered on the axis of rotation and rigidly bound together will be twice the moment of inertia of either object considered separately. [Take two cones, glue them together and the moment of inertia is twice that of either alone].

 

[Buzzdiamond1]

I appreciate your response, but I will respectfully have to disagree. Reason being, it's not like we're taking a cylinder and taking another cylinder, flipping it around 180* on top of it, as all you'd get is a longer cylinder. When you take a single cone, then flip around another cone 180* so the two bases connect, you get a totally different shape, which now would have different properties, similar to taking two rods and forming a cross. Now you would have two different MOI's, one for a rod and one for a cross, correct..?

Another thing is, if you stacked a second cone on top of the first cone, with the base sitting on the point, I'm assuming based on the responses the MOI formula won't change, because all you're doing is changing the mass, am I following the logic correctly..? Now I could believe that, but when 2 cones are stacked this way, it won't spin for very long without toppling over. On the other hand, when you invert one of the cones, stacking both bases together, you get a totally different spin time, spinning much longer, correct..? Therefore, this would lead one to believe the MOI would also be different, unless I'm missing something..?

I'm trying to understand and thank you for your time.

 

[jbriggs444]

I appreciate your response, but I will respectfully have to disagree. Reason being, it's not like we're taking a cylinder and taking another cylinder, flipping it around 180* on top of it, as all you'd get is a longer cylinder. When you take a single cone, then flip around another cone 180* so the two bases connect, you get a totally different shape, which now would have different properties, similar to taking two rods and forming a cross. Now you would have two different MOI's, one for a rod and one for a cross, correct..?

Wrong. That's why I specified a plane of symmetry for the reflection. It's not a rotation. It's a reflection with respect to a particular plane.

Another thing is, if you stacked a second cone on top of the first cone, with the base sitting on the point, I'm assuming based on the responses the MOI formula won't change, because all you're doing is changing the mass, am I following the logic correctly..? Now I could believe that, but when 2 cones are stacked this way, it won't spin for very long without toppling over. On the other hand, when you invert one of the cones, stacking both bases together, you get a totally different spin time, spinning much longer, correct..? Therefore, this would lead one to believe the MOI would also be different, unless I'm missing something..?

Can you recite the definition of moment of inertia for us? This blather leads one to suppose otherwise.

Hint: it does not involve friction or surface areas in contact with the table

 

[Buzzdiamond1]

Yes, it's a quantity expressing a body's tendency to resist angular acceleration.

My understanding of this is basically, when an object is spinning, how much force would it take to move it from a standing position and/or in a change of direction. Am I way off base here with my thinking..?

 

[jbriggs444]

Yes, it's a quantity expressing a body's tendency to resist angular acceleration.

My understanding of this is basically, when an object is spinning, how much force would it take to move it from a standing position and/or in a change of direction. Am I way off base here with my thinking..?

If you change the torque (e.g. by having it spin on its flat base instead of its pointed tip),you change the time taken to spin down without necessarily changing the moment of inertia. So you cannot measure moment of inertia by looking at how long a top takes to spin down without also considering how much torque is applied.

Note that force and torque are two different things.

 

[Buzzdiamond1]

OK, I wasn't looking at it like that. I was thinking that when 2 spinning objects are spinning, the one that spins faster and/or longer, would be harder to change directions than a slower spinning object, as the gyroscopic force would be greater in the object which spins faster/longer.

Maybe I'm mixing the force MOI with the force of Angular Momentum..?

 

[jbriggs444]

TIme to stop spitting out phrases like "faster", "longer", "gyroscopic force", "force MOI" and "force of Angular Momentum", learn some physics and calculate the moment of inertia of a shape... any shape.

 

[Buzzdiamond1]

I'm trying to calculate the MOI of a double diamond, base to base and/or verify the physics study that was already done. I've watched some videos on MOI but it seems like a higher MOI is similar to high RG and low RG, based on how fast objects roll. ,

Sorry for the confusion, but physics is confusing and/or very convoluted with the terminology.

 

[jbriggs444]

I'm trying to calculate the MOI of a double diamond, base to base and/or verify the physics study that was already done. I've watched some videos on MOI but it seems like a higher MOI is similar to high RG and low RG, based on how fast objects roll.

If you want to calculate a moment of inertia, you should, as has already been suggested, calculate a moment of inertia. The way to do that is to learn calculus, not to watch videos about inclined planes.

Let's start with an easy one. Suppose that we have a thin rod of length r and mass m anchored to an axis of rotation at one end and extending outward at right angles to the axis. How would you go about calculating its moment of inertia without referencing a web site?

 

[Buzzdiamond1]

It's already done for me in Dr. Joseph Howard's report, you just don't agree with the calculations and/or his thought process. This is fine, but until I see calculations showing me otherwise, which no one here has provided, as you're asking me to provide you, then this is what I'll have to go on.

Plus, from what I've been able to gather, there's nothing in print stating the MOI of this shape, which is kind of odd to me. Anyways, thank you for your time.

 

[jbriggs444]

It's already done for me in Dr. Joseph Howard's report

Valid reference, please.

Ahh, you seem to believe that the Bowling pin web page mentioned earlier is valid and that the formula on page 3 is validly obtained. However, as I had pointed out in #22, that result would be valid only if M is taken as the mass of a half-cone.

 

[Nugatory]

My understanding of this is basically, when an object is spinning, how much force would it take to move it from a standing position and/or in a change of direction. Am I way off base here with my thinking..?

Yes. The moment of inertia tells you how the spinning speed (think revolutions per minute) of an object changes when you apply a torque to it (try to spin it faster or slower, or rotate it). It has very little to do with moving an object from a standing position or changing its direction if it's already moving.

 

[Buzzdiamond1]

Valid reference, please.

Ahh, you seem to believe that the Bowling pin web page mentioned earlier is valid and that the formula on page 3 is validly obtained. However, as I had pointed out in #22, that result would be valid only if M is taken as the mass of a half-cone.

Dr. Joseph Howard is a physics professor at Salisbury University, why would there be any doubt to the validity of his work, which was done as a class project..? All you're saying is no, it's wrong, without providing any of your work on this subject to substantiate YOUR claim. Seems a bit hypocritical to me..?

Why would a physics professor have the notion to double the value of the formula using symmetry..?

Like I also stated, previously to this report 20 years earlier, another report was loosely done, with the same finding. Seems improbable that 2 separate people would conclude the same finding, with one being a college physics professor to boot..! Smh...

 

[Buzzdiamond1]

Yes. The moment of inertia tells you how the spinning speed (think revolutions per minute) of an object changes when you apply a torque to it (try to spin it faster or slower, or rotate it). It has very little to do with moving an object from a standing position or changing its direction if it's already moving.

I understand what you're saying, but the Radius of Gyration seems to encompass the notion of the spinning speed of an object as well..?

mo·ment of in·er·tia
noun
PHYSICS
noun: moment of inertia; plural noun: moments of inertia

1. a quantity expressing a body's tendency to resist angular acceleration.

The term "angular" in the definition, implies to me that there's a change of direction..? Like if you bump into something, how much resistance is there to the bump, trying to change the spinning body's direction of travel..? That's how I interpret "tendency to resist angular acceleration". Now Radius of Gyration, higher or lower RG numbers tell you specifically how fast or slow something will spin. Having said that, the faster something spins, the higher gyroscopic force is generated, which will stabilize the object, which will now need more force to hit it in order to move it.

So if I'm understanding correctly, the amount of torque required to make the object spin, is what the MOI is called..? I really do think many of these properties are intertwined amongst each other, because the definition of Moment Of Inertia, with the use of the word angular, sure confuses things. :)

 

[jbriggs444]

Dr. Joseph Howard is a physics professor at Salisbury University, why would there be any doubt to the validity of his work, which was done as a class project..? All you're saying is no, it's wrong, without providing any of your work on this subject to substantiate YOUR claim. Seems a bit hypocritical to me..?

I showed my work. It matches the professor's work except that the professor neglected to define his terms. As I explained twice already.

You have shown no work at all.

 

[Buzzdiamond1]

I showed my work. It matches the professor's work except that the professor neglected to define his terms. As I explained twice already.

You have shown no work at all.

I apologize if I missed it, but I haven't seen your work. Maybe the professor neglected to define his terms, because he is fluent with respect to bowling ball design and/or the rotation..? As for my work, I'm not a physics major or professor, so you will not see any formulas from me.

As far as the planes of rotation go, are you saying that if you spin a cone, through the z axis (point through center of base), the MOI is the same if it spins on the base or the point..? Or, if you rotate the cone about the point, vs rotating the cone about the base, the MOI's are the same..?

I understand if the cone rotates about the center of the cone, that would be how the MOI of the cone is determined, but with a double cone, the center point at which the double cone rotates around, is really now the base of a single cone. Because the rotating pints are different between a single cone and a double cone, wouldn't the MOI and/or the formula be different for both..?

 

[jbriggs444]

I haven't seen your work

You replied to it. Angular momentum is additive. The angular momentum of two objects taken together is equal to the sum of their individual angular momenta. [As long as the same axis of rotation is used for all three momenta]. Ergo the moment of inertia of two identical (*) cones glued together is twice the angular momentum of either one alone.

(*) One does require that the two identical cones be positioned identically with respect to the axis.

Maybe the professor neglected to define his terms

Indeed. You tell me, what does the M in 3/5MR^2 denote? The mass of one half of a double cone? Or of the whole thing?

As far as the planes of rotation go, are you saying that if you spin a cone, through the z axis (point through center of base), the MOI is the same if it spins on the base or the point..? Or, if you rotate the cone about the point, vs rotating the cone about the base, the MOI's are the same..?

Edit: Re-reading your question "spins on the base versus spins on the point". Those are both rotations about the same axis. Moment of inertia will be identical.

I understand if the cone rotates about the center of the cone, that would be how the MOI of the cone is determined, but with a double cone, the center point at which the double cone rotates around, is really now the base of a single cone. Because the rotating pints are different between a single cone and a double cone, wouldn't the MOI and/or the formula be different for both..?

If one is talking about a "moment of inertia" then one is talking about rotations in a plane -- about an axis which is a line, not a point. It does not matter whether one identifies this line with the point of one cone, the point of the other cone or the center of the flat part where the two cones are joined. All three points fall on the same axis of rotation.

Given an axis of rotation, the moment of inertia of an object is defined as an integral taken over the volume of the object being evaluated. This is the integral of each incremental volume element multiplied by its mass density times the square of the distance of that volume element from the center of rotation. From this definition it is patently obvious that the moment of inertia of two objects taken together is the sum of their individual moments of inertia and that the moment of inertia of an object reflected about a plane at right angles to the axis is identical to that of the unreflected object.

Edit to add: It is also obvious that moving the object up or down the axis will leave its moment of inertia unchanged.

 

[Buzzdiamond1]

You replied to it. Angular momentum is additive. The angular momentum of two objects taken together is equal to the sum of their individual angular momenta. [As long as the same axis of rotation is used for all three momenta]. Ergo the moment of inertia of two identical (*) cones glued together is twice the angular momentum of either one alone.

(*) One does require that the two identical cones be positioned identically with respect to the axis.

I'm sorry sir, but I'm not getting what you mean by "your work"..? Please point out what number I replied to, in reference to your work..?

OK, so if you stack/glue two balls on top of each other, spinning it like a top, are you saying the moment of inertia will be twice the angular momentum of either one alone..?

Indeed. You tell me, what does the M in 3/5MR^2 denote? The mass of one half of a double cone? Or of the whole thing?

My interpretation is that he was referring to the mass of the entire double cone.

Edit: Re-reading your question "spins on the base versus spins on the point". Those are both rotations about the same axis. Moment of inertia will be identical.

So if you put two cones together, one way point to point(like an hourglass) and another way base to base(like the diamond I'm referring to), are you telling me both will have the same MOI..? Won't one rotate faster than the other, similar to that of a hollow cylinder and a solid cylinder, therefore, the MOI's will also be different..?

If one is talking about a "moment of inertia" then one is talking about rotations in a plane -- about an axis which is a line, not a point. It does not matter whether one identifies this line with the point of one cone, the point of the other cone or the center of the flat part where the two cones are joined. All three points fall on the same axis of rotation.

Given an axis of rotation, the moment of inertia of an object is defined as an integral taken over the volume of the object being evaluated. This is the integral of each incremental volume element multiplied by its mass density times the square of the distance of that volume element from the center of rotation. From this definition it is patently obvious that the moment of inertia of two objects taken together is the sum of their individual moments of inertia and that the moment of inertia of an object reflected about a plane at right angles to the axis is identical to that of the unreflected object.

Edit to add: It is also obvious that moving the object up or down the axis will leave its moment of inertia unchanged.

I was referring to a line going from the point of the cone, down through the center of the base.

 

[Nidum]

So if you put two cones together, one way point to point(like an hourglass) and another way base to base(like the diamond I'm referring to), are you telling me both will have the same MOI..? Won't one rotate faster than the other, similar to that of a hollow cylinder and a solid cylinder, therefore, the MOI's will also be different..?

(1) Both configurations have the same MoI .

(2) For any number of objects assembled on a common axis then the MoI of the whole is the sum of the individual object MoI's .

(3) Objects need not be same size , geometry or density ¹.

(4) When working out the MoI of an object with complex geometry the standard method is to break the object down into subsections with simpler geometry , work out the MoI's of the subsections and then sum them all to get the total MoI .

For objects where the geometry can be defined by equations calculus methods can be used . Otherwise numerical methods are used .

(5) This is not all theoretical . When working out the MoI's of practical components in engineering design work these methods are used daily . Traditionally done by hand but now commonly done using CAD systems - method is just the same though .Note 1 : There has to be an adequate torque connection between the objects and the assembly has to be stiff enough not to change shape significantly as it rotates .

 

[Buzzdiamond1]

(1) Both configurations have the same MoI .

(2) For any number of objects assembled on a common axis then the MoI of the whole is the sum of the individual object MoI's .

(3) Objects need not be same size , geometry or density ¹.

(4) When working out the MoI of an object with complex geometry the standard method is to break the object down into subsections with simpler geometry , work out the MoI's of the subsections and then sum them all to get the total MoI .

For objects where the geometry can be defined by equations calculus methods can be used . Otherwise numerical methods are used .

(5) This is not all theoretical . When working out the MoI's of practical components in engineering design work these methods are used daily . Traditionally done by hand but now commonly done using CAD systems - method is just the same though .Note 1 : There has to be an adequate torque connection between the objects and the assembly has to be stiff enough not to change shape significantly as it rotates .

1. You and a lot of other people are saying both configurations have the same MOI and/or use the same MOI formula, only because someone interpreted the formula at one time, plugged it into the CAD programs and now that's what everybody goes by. But, if that original person interpreted things wrong, then the formula now used is incorrect, correct..? What proof do we have that the formula for a double cone, base to base like a diamond, is actually I = 3/10MR^2..? It isn't published anywhere, now is it..?

2. Yes, a number of objects assembled together on a common axis is the sum of both put together, because the basic shape doesn't change, it only get taller, longer or wider, but retaining the same basic shape.

Now I can also see where a shape is split in half, like a sphere and a hemisphere, the same formula is used for the hemisphere (half of the whole), as the formula for a hemisphere was derived FIRST from the whole. But, in the case of a base to base cone and/or a diamond, the whole was derived from the half. There's a big difference there. Do you get my point..?

 

[jbriggs444]

1. You and a lot of other people are saying both configurations have the same MOI and/or use the same MOI formula, only because someone interpreted the formula at one time, plugged it into the CAD programs and now that's what everybody goes by.

Do the integration yourself. Otherwise you have no grounds to argue.

 

[berkeman]

Do the integration yourself. Otherwise you have no grounds to argue.

Yes, we are done wasting folks' time here. Please do the integral and send me your work in a Private Message, and I'll re-open this thread so you can post it here for people to check. Thread is closed for now.