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Inertia
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Inertia is the resistance an object has to a change in its state of motion. This tendency
varies with mass. Mass is that quantity that is solely dependent upon the inertia of an
object. The more inertia that an object has, the more mass that it has. A more massive
object has a greater tendency to resist changes in its state of motion. Newton's first
law that states that every object in a state of uniform motion tends to remain in that state
of motion unless an external force is applied to it, this is often referred to as the Law of
Inertia.
Moment of inertia is the name given to rotational inertia, the rotational analog of mass for
linear motion. The moment of inertia must be specified with respect to a chosen axis of
rotation. For a point mass the moment of inertia is just the mass times the square of
perpendicular distance to the rotation axis, I = mr^{2}. That point mass relationship
becomes the basis for all other moments of inertia since any object can be built up
from a collection of point masses.
r
+---------------------- m (point mass)
Axis of rotation
I = mr^{2}
dI = r^{2}∫dm
Example 1. Cylinder
dm = ρdV
= 2πrLρdr
_{R}
I = 2πρL∫r^{3}dr
^{0}
= 2πρLR^{4}/4
= 2πMLR^{4}/4πR^{2}L
= (1/2)mR^{2}