Let's calculate the moment of inertia for a rotating rod. The general definition for the moment of inertia is I = f r²dm Let's suppose we have a uniform rod of length R with mass M, then our line density is M where we have ignored the thickness of the rod to simplify the problem. Density is an intrinsic variable, meaning that it is a constant and doesn't change for varying lengths; thus we can rephrase this equation in terms of the measured infinitesimals as dm = A dr As a reminder, the r in our moment of inertia definition is the distance from the rotating axis to the mass, so rotating about the edge of rod will yield a different result than if we rotated about the middle of the rod. Let's do the latter example here and save the former as an exercise for you to do. If we rotate the rod through the center, then our integration limits become -R/2 to R/2, S+R/2 |+R/2
Let's calculate the moment of inertia for a rotating rod. The general definition for the moment of inertia is I = f r²dm Let's suppose we have a uniform rod of length R with mass M, then our line density is M where we have ignored the thickness of the rod to simplify the problem. Density is an intrinsic variable, meaning that it is a constant and doesn't change for varying lengths; thus we can rephrase this equation in terms of the measured infinitesimals as dm = A dr As a reminder, the r in our moment of inertia definition is the distance from the rotating axis to the mass, so rotating about the edge of rod will yield a different result than if we rotated about the middle of the rod. Let's do the latter example here and save the former as an exercise for you to do. If we rotate the rod through the center, then our integration limits become -R/2 to R/2, S+R/2 |+R/2
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Rigid Body
A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.
Rigid Body Dynamics
Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).
Question
Transcribed Image Text:Let's calculate the moment of inertia for a rotating
rod. The general definition for the moment of inertia
is
I = fr²dm
Let's suppose we have a uniform rod of length R with
mass M, then our line density is
M
where we have ignored the thickness of the rod to
simplify the problem.
Density is an intrinsic variable, meaning that it is a
constant and doesn't change for varying lengths; thus
we can rephrase this equation in terms of the
measured infinitesimals as
dm = A dr
%3D
As a reminder, the r in our moment of inertia
definition is the distance from the rotating axis to the
mass, so rotating about the edge of rod will yield a
different result than if we rotated about the middle of
the rod. Let's do the latter example here and save the
former as an exercise for you to do.
If we rotate the rod through the center, then our
integration limits become -R/2 to R/2,
+R/2
|+R/2
1
dr =
1
AR3
12
Irod
R/2
-R/2
Transcribed Image Text:In the example as above, we were rotating the rod
about its midpoint with the rod perpendicular to the
rotation axis. Now as an exercise, suppose we shift
the rotation point to the end of the rod. Now the rod
is rotating about an end and perpendicular to the
rotation axis. What should be the integral for the
moment of inertia of the rod?
R/2
Irod
-R/2
R/2
Irod = 2
%3D
0.
R
Irod
Irod
R
none of the above
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