Any body cannot instantly change the speed. This property is called inertness. For progressively driving body, a measure of inertness is weight, and for rotating – the inertia moment which depends on weight, a form and an axis around which the body moves. Therefore there is no uniform formula for measurement of the moment of inertia, for each body it is the.
It is required to you
- - the mass of the rotating bodies;
- - tool for measurement of radiuses.
1. For calculation of the moment of inertia for any body, take integral from function which represents the distance square from an axis depending on distribution of weight in dependence removal of r from it? dm. As to take such integral very difficult, correlate a body which moment of inertia is calculated to for what this size is already calculated.
2. For bodies which have the correct formula use Steiner's theorem considering passing of an axis of rotation through a body. For each of bodies calculate the inertia moment by a formula which is received from the corresponding theorem.
3. For a continuous core the mass of m which axis of rotation passes through one of its ends, I=1/3·m· l?, where l is length of a continuous core. If the axis of rotation of a core passes through the middle of such core, then its moment to inertia is equal to I=1/12·m· l?.
4. If the material point rotates around a motionless axis (model of orbital rotation), then to find its moment of inertia increase its mass of m by a square of radius of rotation of r (I=m • r?). The same formula is used for calculation of the moment of inertia of a thin hoop. Calculate the moment of inertia of a disk which is equal to I=1/2·m· r? less moment of inertia of a hoop due to uniform distribution of weight on all body. On the same formula calculate the inertia moment for a continuous disk.
5. To calculate the inertia moment for the sphere, increase its mass of m by a square of radius of r and coefficient 2/3 (I=2/3·m· r?). For a sphere r radius from substance which mass is distributed evenly and equal to m calculate the inertia moment by formula I=2/5 • m · r?.
6. If the sphere and a sphere have the identical weight and radius, then the moment of inertia of a sphere due to uniform distribution of weight is less, than at the sphere which mass is dispersed on an outer sheath. Considering the inertia moment, calculate dynamics of rotary motion and kinetic energy of rotary motion.