How do we compute the moment of inertia matrix of a thin square plate of mass M with sides of size a?
the moment of inertia is dependent on the axis of rotation
the plate has different momenta due to its shape
google moment of inertia
and get lots of examples for various shapes.
Hi oobleck, just wanted a clarification on what they've meant here by "moment of inertia matrix".
Thanks!
I only post a question once I've googled and found something which I don't understand :-)
my google of "moment of inertia matrix" tuned up half a dozen hits on the first page...
You do not say about what axis or if around the center of mass. If not at center of mass and about principal axis then much more difficult (like dynamically unbalanced car wheel) Google inertia tensor .
To compute the moment of inertia matrix of a thin square plate, we need to consider the rotational inertia around each of the principal axes of the plate. The moment of inertia matrix provides information about these rotational inertias with respect to each axis.
For a thin square plate with sides of size "a" and a mass "M", the moment of inertia matrix can be computed as follows:
1. Start by considering the moment of inertia around the x-axis. This represents the rotational inertia when the plate is rotated around an axis parallel to one of its sides.
- The moment of inertia around the x-axis, denoted as Ixx, can be calculated using the formula: Ixx = (M * (a^2)) / 12.
2. Next, calculate the moment of inertia around the y-axis. This represents the rotational inertia when the plate is rotated around an axis perpendicular to its sides.
- The moment of inertia around the y-axis, denoted as Iyy, is also given by Iyy = (M * (a^2)) / 12.
3. Finally, compute the moment of inertia around the z-axis. This represents the rotational inertia when the plate is rotated around an axis perpendicular to the plate.
- The moment of inertia around the z-axis, denoted as Izz, can be obtained using the formula: Izz = (M * (a^2)) / 6.
After performing these calculations, the moment of inertia matrix for the thin square plate can be written as:
I = | Ixx 0 0 |
| 0 Iyy 0 |
| 0 0 Izz |
Therefore, for a thin square plate with sides of size "a" and mass "M", the moment of inertia matrix would be a diagonal matrix with Ixx, Iyy, and Izz as described above.