Calculus Volume 3 · free preview
Calculating Centers of Mass and Moments of Inertia
Centers of mass divide the first moments by the total mass, and moments of inertia integrate the squared distance times density, the angular motion's resistance, both computed by the triple and double integrals over the solid or lamina.
1. Centers of mass
Centers of mass
2. Moments of inertia
Moments of inertia
3. Computations
Computations
4. The Pappus connection
The Pappus connection
5. Center of mass formula
The x coordinate of the center of mass equals which ratio of integrals?
6. Moment of inertia definition
The moment of inertia about an axis is which integral?
7. Uniform sphere moment
The uniform solid sphere's moment of inertia about a diameter equals which classical value?
8. Parallel axis theorem
The parallel axis theorem states I equals I_cm plus which term?
9. Why squared distance
The moment of inertia weights by the squared distance because of which physical fact?
10. First moment distance
The first moments weight by which power of the distance? Answer with a number.
11. Radius of gyration formula
The radius of gyration k equals the root of I over which letter? Answer with one lowercase letter.
12. Hemisphere center location
The uniform hemisphere's center of mass lies on its axis of symmetry. As a fraction of the radius, its height is 3a over what number? Answer with a number.
13. Symmetry shortcut
The center of mass of a uniform body with a plane of symmetry lies where?
14. Disk moment coefficient
The uniform disk's moment about its center is 1/2 M a squared. As a fraction, what is the coefficient? Answer like 1/2.
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