Momentsof Inertia - Lecture notes 6 PDF

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Moments of Inertia...

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Moments of Inertia for Ring, Disc and Sphere

Ring of mass M and radius R All mass is distance R away from the center. Axis of rotation passes through the center perpendicularly to the disc: ∫

∫

∫

Disc (thin) of mass M and radius R Disc can be thought of as a collection of rings with radii going from 0 to R. Then to get the moment inertia of the disc, we sum up (integrate over) the moments of inertia of all the rings: ∫ .

For a disc, mass is distributed over its surface; so, the mass density is Mass/Surface Area, which gives: . Infinitesimal surface area of the disc is , which is circumference of a ring multiplied by its thickness , so that

∫

∫

∫

. Therefore:

∫

Solid cylinder of mass M, radius R and length L Solid cylinder can be thought of as a collection of discs of specific thickness with radii going from 0 to R, stacked along the z-axis. Then to get the moment inertia of the cylinder, we sum up (integrate over) the moments of inertia of all the discs: ∫ .

For a cylinder of finite thickness, mass is distributed inside its volume; so, the mass density is Mass/Volume. Since these discs comprise the cylinder Mass/Volume can be found as: (total mass of the cylinder divided by the total volume of the cylinder). Infinitesimal volume of the cylinder is , which is surface area of the disc multiplied by its thickness , so that . Therefore: ∫

∫

∫

∫

Solid Sphere of mass M and radius R Solid sphere can be thought of as a collection of discs of specific thickness with radii going from 0 to R, stacked along the z -axis. Then, to get the moment inertia of the sphere, we sum up (integrate over) the moments of inertia of all the discs: ∫ .

For a disc of finite thickness, mass is distributed inside its volume; so, the mass density is Mass/Volume. Since these discs comprise the solid sphere Mass/Volume can be found as: (total mass of the sphere divided by the total volume of the sphere). Infinitesimal volume of the disc is , which is surface area of the disc multiplied by its thickness , so that . Therefore: ∫

∫

∫

∫

Now, coordinates r and z are not unrelated; they relate as

∫

∫

,

, so that...