Moment of inertia
DOWNLOAD
FREE
|
Addition law for moments of inertia for two or more particles rotating about the same axis of rotati
Consider a system of n particles of possibly differing mass Mi and distance ri (where 1 = i = n) from a fixed axis or rotation, L. If all the particles in the system are rotating together then the moment of inertia of the whole system is given by: [Equation goes here - download the original to see it.] In other words, for particles we simply add the moments of inertia to each particle to find the moment of inertia of a composite body. We will demonstrate the validity of this result for a composite body of two particles. [Diagram goes here - download the original to see it.] Let P and Q be two particles of mass m1 and m2 and perpendicular distance r1 and r2 from an axis of rotation L. The total torque applied to this system is: [Equation goes here - download the original to see it.] where FP and FQ are the tangental components of the forces acting on P and Q respectively. Both particles are rigidly "fixed" to each other in someway, so their angular acceleration is the same: [Equation goes here - download the original to see it.] The moment of inertia of each particle is: [Equation goes here - download the original to see it.] Let I = total momentum. Then applying Newton's Second law in the form [Equation goes here - download the original to see it.] The full result for n particles would follow by mathematical induction. We will now illustrate an application of this result. Example (1) Particles of mass m, 3m and 5m are situated at points (-2, 1), (2, 3) and (4, -1) respectively from O in the x, y plane. Find the moment of inertia of the whole system about the z-axis. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation 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
|
