Moment of inertia
DOWNLOAD
FREE
|
Standard results for moments of inertia
We now proceed to use the result [Equation goes here - download the original to see it.] to derive the standard results for the moment of inertia of a rod. Determination of these standard results by integration from first principals is, however, not required by the syllabus, and these derivations can be omitted. The student may choose to accept the results as standard results and proceed to the next part of the theory. Rod Rod of length 2l and mass M rotating about an axis perpendicular to the rod through the centre of mass has moment of inertia [Equation goes here - download the original to see it.] Proof [Diagram goes here - download the original to see it.] We divide the rod into segments each of length dx. Let ? be the density of the rod - that is here the mass per unit length. We treat each element as a particle of mass M, given by [Equation goes here - download the original to see it.] Each segment is approximately a particle of mass M, and hence of moment inertia [Equation goes here - download the original to see it.] where x is the distance of the segment from the axis of the rotation. Then the moment of inertia of the rod as a whole is given by [Equation goes here - download the original to see it.] And in the limit, as dx ? 0, the approximation becomes exact: [Equation goes here - download the original to see it.] Now the total mass is given by: [Equation goes here - download the original to see it.] In the limit, when dx ? 0 [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Shown The other standard results are proven similarly. The standard results are as follows. [Table goes here - download the original to see it.]
|
Contents of
Moment of inertia
1 Moment of Inertia
2 Definition of inertia for a particle
3 Addition law for moments of inertia for two or more particles rotating about the same axis of rotati
4 Standard results for moments of inertia
|
