# Finding the product moment of inertia of this cylinder

• anchonee
In summary, the question is about the value of Iyz and whether it is equal to 0. Some solutions suggest that it is, reasoning that the geometry is symmetrical. However, upon closer examination of the YZ plane, it is not symmetrical about either axis. Another solution suggests using the parallel axis theorem, taking the angular momentum from the cylinder's mass centre. However, when trying to take it from point O, the answer does not match the correct one. The issue is that the rotation in the j direction is not about an axis through O, so the correct answer only includes the Iyy term.
anchonee
So the following question is attached (There is another thread with the same question but no solution to what I am asking on there)

Now according to several solutions, apparently IYZ is equal to 0, and they reason this by saying that the geometry is symmetrical.

However when looking at the YZ plane, the geometry is not symmetrical about either axis, so therefore ∫yz.dm ≠ 0?

anchonee said:
So the following question is attached (There is another thread with the same question but no solution to what I am asking on there)
View attachment 84362
View attachment 84363

Now according to several solutions, apparently IYZ is equal to 0, and they reason this by saying that the geometry is symmetrical.

However when looking at the YZ plane, the geometry is not symmetrical about either axis, so therefore ∫yz.dm ≠ 0?
As I posted on the other thread, it may be that Iyz is being used to refer to the MoI about the cylinder's mass centre. The parallel axis theorem can then be used in conjunction.

anchonee
haruspex said:
As I posted on the other thread, it may be that Iyz is being used to refer to the MoI about the cylinder's mass centre. The parallel axis theorem can then be used in conjunction.

I have tried the question by taking the angular momentum from the centre of mass, then using the parallel axis theorem.
e.g. I used Ho = Hg + (r x G)

That works perfectly fine, the answer was correct. But why is it not the case when I try to take it from point O?

Okay let's say for example I take it from point O.

ωx = ωi
ωy = pj
ωz = 0k

Therefore you would have the following for Ho:
Ho = (-Ixx - Iyx - Izxx i + (-Ixy + Iyy - Izyy j

From point O,
Ixoxo = IG + md2 (parallel axis theorem) = (1/4)mr2 + (1/3)mb2 + mh2
Due to symmetry,
Iyozo = Izoxo = 0
Iyoyo = (1/2)mr2

Now technically from point O, the mass is not symmetrical about either axis in the YZ plane.
I would have taken it as Izoyo = (-b/2)×h×m
This appears to also be valid when I use Izy = ∫zy.dm

Therefore that would result in my answer being:

Ho = mω(r2/4 + b2/3 + h2) i + ((1/2)mr2 - (b/2)×h×m)p j

Ho = mω(r2/4 + b2/3 + h2) i + ((1/2)mr2)p j

..any ideas?

Last edited:
The p rotation is not about an axis through O.

anchonee
haruspex said:
The p rotation is not about an axis through O.
So how would it change what I did in the above methodology?

anchonee said:
So how would it change what I did in the above methodology?
In the j direction, the axis of rotation is through the mass centre, so there is only an ##Iyy## term.

anchonee
haruspex said:
In the j direction, the axis of rotation is through the mass centre, so there is only an ##Iyy## term.
perfect. thank you!

## 1. What is the product moment of inertia of a cylinder?

The product moment of inertia, or simply moment of inertia, of a cylinder is a measure of its resistance to changes in its rotational motion. It is calculated by multiplying the mass of the cylinder by the square of its distance from the axis of rotation.

## 2. How do you find the product moment of inertia of a cylinder?

To find the product moment of inertia of a cylinder, you need to know the mass of the cylinder and the distance between the axis of rotation and the center of mass. Then, you can use the formula I = mr^2, where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation to the center of mass.

## 3. What is the equation for calculating moment of inertia?

The equation for calculating moment of inertia is I = mr^2, where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation to the center of mass. This equation applies to all objects, including cylinders.

## 4. How does the shape of a cylinder affect its moment of inertia?

The shape of a cylinder does not affect its moment of inertia as long as the mass and distance from the axis of rotation are the same. This means that a hollow cylinder and a solid cylinder with the same mass and dimensions will have the same moment of inertia.

## 5. Why is it important to find the product moment of inertia of a cylinder?

Finding the product moment of inertia of a cylinder is important because it helps us understand the rotational motion of the object. It is also necessary for calculating the torque, or rotational force, applied to the cylinder, as well as its angular acceleration. This information is crucial in many engineering and physics applications.

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