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

[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.