Parallel-axis theorem

Definition

The parallel-axis theorem allows the calculation of a body's moment of inertia about any axis, given its moment of inertia about a parallel axis through the object's center of mass and the perpendicular distance between the axes. It is expressed as $I = I_{\text{cm}} + Md^2$, where $I$ is the moment of inertia about the new axis, $I_{\text{cm}}$ is the moment of inertia about the center-of-mass axis, $M$ is the mass of the body, and $d$ is the distance between the two axes.

5 Must Know Facts For Your Next Test

1. The parallel-axis theorem applies to rigid bodies and aids in calculating moments of inertia for different rotational axes.
2. The formula for the theorem is $I = I_{\text{cm}} + Md^2$. The terms represent: $I$ (moment of inertia about a new axis), $I_{\text{cm}}$ (moment of inertia about center-of-mass axis), $M$ (mass), and $d$ (distance between axes).
3. This theorem simplifies finding moments of inertia without directly integrating over a new rotational axis.
4. It is crucial for analyzing systems with shifted rotational points like pendulums or rotating machinery components.
5. Understanding this concept requires familiarity with basic moments of inertia for standard shapes such as rods, discs, and spheres.

Review Questions

• What does each term represent in the equation $I = I_{\text{cm}} + Md^2$?
• How would you use the parallel-axis theorem to find the moment of inertia for a rod rotated around one end?
• Why is it important to know both the center-of-mass moment of inertia and the distance between axes in using this theorem?

Related terms

Moment of Inertia: A measure of an object's resistance to changes in its rotation rate. It depends on both mass distribution and axis location.

Center of Mass: A point representing the mean position of matter in a body or system. It's where mass can be considered concentrated for analysis.

Radius of Gyration: $k$, which represents how far from an axis all mass could be concentrated without changing rotational characteristics.