- #1
sparkle123
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Moment of inertia: vector derivation
- Thread starter sparkle123
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In summary, we discussed the representation of torque and its components. The matrix was obtained by multiplying the vector r with the vector α, and then splitting off α to create a 3x3 matrix. We also learned about the concept of r^2-matrix and how it works.
- #2
I like Serena
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Hey sparkle! 
Did you try to write out the 3rd line?
$$mr^2 \mathbf{α} = (r^2 I) \mathbf{α}$$
$$(\mathbf{r} \cdot \mathbf{α})\mathbf{r} = ([x\ y\ z] \cdot \mathbf{α}) \begin{bmatrix}x \\ y \\ z \end{bmatrix} = ([x\ y\ z] \cdot \begin{bmatrix}x \\ y \\ z \end{bmatrix})\mathbf{α}$$
(You can check that last equality by writing it out in components.)
Did you try to write out the 3rd line?
$$mr^2 \mathbf{α} = (r^2 I) \mathbf{α}$$
$$(\mathbf{r} \cdot \mathbf{α})\mathbf{r} = ([x\ y\ z] \cdot \mathbf{α}) \begin{bmatrix}x \\ y \\ z \end{bmatrix} = ([x\ y\ z] \cdot \begin{bmatrix}x \\ y \\ z \end{bmatrix})\mathbf{α}$$
(You can check that last equality by writing it out in components.)
- #3
sparkle123
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Hi I like Serena!
Do we get:
$$ (\mathbf{r} \cdot \mathbf{α})\mathbf{r} = ([x\ y\ z] \cdot \begin{bmatrix}x \\ y \\ z \end{bmatrix})\mathbf{α} = (x^2 + y^2 + z^2)\mathbf{α}$$
$$∴ mr^2 \mathbf{α} - m(\mathbf{r} \cdot \mathbf{α})\mathbf{r} = m(r^2 - x^2 - y^2 - z^2)\mathbf{α}$$
I still don't see how we get the 3X3 matrix
Thanks again!
Do we get:
$$ (\mathbf{r} \cdot \mathbf{α})\mathbf{r} = ([x\ y\ z] \cdot \begin{bmatrix}x \\ y \\ z \end{bmatrix})\mathbf{α} = (x^2 + y^2 + z^2)\mathbf{α}$$
$$∴ mr^2 \mathbf{α} - m(\mathbf{r} \cdot \mathbf{α})\mathbf{r} = m(r^2 - x^2 - y^2 - z^2)\mathbf{α}$$
I still don't see how we get the 3X3 matrix
Thanks again!
- #4
I like Serena
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I have to admit that my representation was not correct. Sorry.
Let's write it out in its components.
$$ (\mathbf{r} \cdot \mathbf{α})\mathbf{r} = (x α_x + y α_y + z α_z ) \begin{bmatrix}x \\ y \\ z \end{bmatrix}=...$$
Can you turn this into a vector without a factor in front?
And then split off ##\mathbf{α}## yielding a matrix?
Let's write it out in its components.
$$ (\mathbf{r} \cdot \mathbf{α})\mathbf{r} = (x α_x + y α_y + z α_z ) \begin{bmatrix}x \\ y \\ z \end{bmatrix}=...$$
Can you turn this into a vector without a factor in front?
And then split off ##\mathbf{α}## yielding a matrix?
- #5
sparkle123
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Okay, I got
$$\left(
\begin{array}{ccc}
x^2 & xy & xz \\
xy & y^2 & yz \\
xz & yz & z^2 \end{array}
\right)\mathbf{α}$$
How does $$r^2 - matrix$$ work?
Thanks! :)
$$\left(
\begin{array}{ccc}
x^2 & xy & xz \\
xy & y^2 & yz \\
xz & yz & z^2 \end{array}
\right)\mathbf{α}$$
How does $$r^2 - matrix$$ work?
Thanks! :)
- #6
I like Serena
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sparkle123 said:
It's
$$r^2 \mathbf{α} - matrix \mathbf{α}$$
and
$$r^2 \mathbf{α} = r^2 \begin{pmatrix} 1 & 0 & 0 \\ 0 &1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \mathbf{α} = \begin{pmatrix} r^2 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \end{pmatrix} \mathbf{α}$$
- #7
sparkle123
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Oh thank you! I should brush up on matrices hehe :)
FAQ: Moment of inertia: vector derivation
What is moment of inertia?
Moment of inertia, also known as rotational inertia, is a measure of an object's resistance to changes in its rotational motion. It is the property of an object that determines how difficult it is to change its rotation about a given axis.
2. How is moment of inertia calculated?
The moment of inertia of an object can be calculated by multiplying the mass of the object by the square of its distance from the axis of rotation. This is represented by the equation I = mr^2, where I is the moment of inertia, m is the mass, and r is the distance from the axis.
3. What is the vector derivation of moment of inertia?
The vector derivation of moment of inertia involves using vector operations, such as cross products, to determine the moment of inertia of a three-dimensional object. This is often used for more complex objects with irregular shapes.
4. Why is moment of inertia important?
Moment of inertia is important because it helps us understand and predict how objects will behave when they are rotating. It is also a key factor in the laws of rotational motion and is used in many engineering and physics applications, such as designing vehicles and machinery.
5. How does moment of inertia differ from mass?
Moment of inertia and mass are two different properties of an object. Mass is a measure of an object's resistance to changes in its linear motion, while moment of inertia is a measure of its resistance to changes in its rotational motion. They are related, but not the same, as an object's mass will affect its moment of inertia, but they are calculated differently.