College Physics II – Mechanics, Sound, Oscillations, and Waves
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.
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The parallel-axis theorem applies to rigid bodies and aids in calculating moments of inertia for different rotational axes.
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).
This theorem simplifies finding moments of inertia without directly integrating over a new rotational axis.
It is crucial for analyzing systems with shifted rotational points like pendulums or rotating machinery components.
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?