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?