Mr.p: Good morning. Today, we are going to
discuss six different equations for moments of inertia of rigid objects with constant density.
"Flipping Physics" Intro
A thin, hollow cylinder about or around its long cylindrical axis;
a solid cylinder about its long cylindrical axis;
a thin hollow sphere about its center of mass (it's a racquetball);
a solid sphere about its center of mass (it's a lacrosse ball);
a thin rod about its center of mass;
and a thin rod about one end.
Again, these are rigid objects with constant density.
Rigid means the objects will not easily change shape.
Billy: Rigid objects do not easily change shape. Okay. These are rigid objects with constant density.
Mr.p: In a typical, calculus-based physics class we would derive many of these equations for moments of inertia,
however, for this algebra-based physics class, we will only discuss their relative moments of inertia.
In my opinion it is not worth memorizing these equations,
however, understanding why they have their relative values is very worthwhile.
Bobby, what is the equation for the moment of inertia of a system of particles?
Bobby: The moment of inertia of a system of particles equals
the mass of each particle times the square of the distance each particle is from the axis of rotation.
Actually, the sum of each of those expressions; one for each particle.
But we are not talking about systems of particles.
We are talking about the moments of inertia of rigid objects with shape and constant density.
Can we use that equation?
Mr.p: Right. So,
in order to derive the equations for moments of inertia of rigid objects with shape and constant density
we would need to use calculus. The equation we would use is moment of inertia equals the integral with respect
to mass of the square of the distance each piece of the object is from the axis of rotation.
However, this is not a calculus-based class, so we are not going to use that equation.
We are instead going to use the algebraic equation for the moment of inertia of a system of particles
to estimate and understand the equations for moments of inertia or rigid objects with constant density. Okay?
BBB: Okay.
Mr.p: Okay. Let's start with the last two: a thin rod about its center of mass and then about one end.
Realize the word "thin" in "thin rod" means the radius of the rod is very small relative to the length of the rod.
We consider the rod to essentially be one-dimensional.
Both moments of inertia for the thin rod are a fraction times ML squared.
Where "M" is the mass of the rod and "L" is the length of the rod.
For example, the moment of inertia of a thin rod rotating like this about its center of mass is
one twelfth the mass of the rod times the length of the rod squared.
Billy, if we instead rotate the rod about one of its ends
Would you expect the moment of inertia to increase decrease or stay the same
relative to rotating the rod about its center of mass? And when you do this, consider small pieces which
make up the rod and how those are affected by the moment of inertia of a system of particles equation.
Billy: Well, moment of inertia is linearly proportional to the mass of each piece of the rod,
but it is proportional to the square of the distance each piece of the rod is from the axis of rotation.
So, when we move the axis of rotation to the end, some of the mass is now farther from the axis of rotation.
Therefore, I would think the moment of inertia would go up, right?
Mr.p: Correct Billy. The moment of inertia of the thin rod about one end is one third mass times length squared
and one third is greater than one twelfth, therefore the
moment of inertia increases when we move the axis of rotation from the center of mass to the end of the rod.
Next let's talk about the other four moments of inertia. Where each is a fraction times M R squared.
Where "M" again is the mass of the object and "R" is the radius of the object.
Let's start with the thin, hollow cylinder about its long cylindrical axis.
Again, the word "thin" here means
the thickness of the hollow cylinder is very small compared to the radius of the cylinder,
therefore we can consider the hollow cylinder to be essentially a two-dimensional object
where every piece of the thin hollow cylinder is a distance R from the axis of rotation.
Bo, what do you think
that means for the moment of inertia of the thin, hollow cylinder about its long cylindrical axis?
Bo: What is a long cylindrical axis?
Bobby: And why are all the letters capitalized?
Mr.p: Oh right.
The long cylindrical axis is the axis of the cylinder which intersects
with the center of every circle created by the cylinder.
In other words, we are rotating the cylinder like this.
We cannot just say, "about its center of mass" because then we could be rotating it this way, for example.
And mass, length, and radius are all capitalized because they refer to the entire object,
rather than a piece of the object.
For example, when we have a radius which varies from zero to capital R,
that radius variable will be denoted using a lowercase r
and the full radius of the object will be denoted using a capital R.
Again Bo,
what do you think we know about the moment of inertia of a thin hollow cylinder about its long cylindrical axis?
Bo: I don't...
Billy: Well, every piece of the cylinder is a distance R, from the axis of rotation...
Bo: ...yeah.
Bobby: So the moment of inertia every piece of the cylinder is the mass of each piece times R squared.
Bobby: Okay, so,
because every piece of the thin hollow cylinder is a distance R from the axis of rotation,
the moment of inertia of the entire thin, hollow cylinder equals
the mass of the cylinder times the square of the radius of the cylinder.
That makes sense. Thanks guys.
Billy: You are welcome.
Bobby: Yeah. But what is the fraction?
Billy: Yeah. What is the fraction in that equation?
Bo: It's 1.
Billy: 1 is not a faction.
Bo: Fine, it's 1 over 1.
Billy: Is 1 over 1 a fraction?
Mr.p: Great. So now let's move on to the moment of inertia a solid cylinder
again about its long cylindrical axis.
Bobby, how do you think
the fraction multiplied by the mass times radius squared for the moment of inertia of a solid cylinder
about its long cylindrical axis compares to the fraction for a thin hollow cylinder?
Bobby: Okay.
So this comes down to where the mass of the solid cylinder is located relative to the axis of rotation
Because the the cylinder is solid, more of the mass of the solid cylinder is closer to the axis of rotation
than for the thin, hollow cylinder. So the moment of inertia should be less,
therefore the fraction for the solid cylinder should be less than the fraction for the thin, hollow cylinder.
Mr.p: Correct.
The moment of inertia of a solid cylinder about its long cylindrical axis is one half mass times radius squared.
One half is less than 1, so the fraction
for the moment of inertia of a rigid, solid cylinder of constant density about its long cylindrical axis
is less than the fraction for the moment of inertia of a thin, hollow cylinder.
Now, notice neither of these two moment of inertia equations depend on the length of the cylinder.
In other words, the length of the cylinder is irrelevant
as far as the equations for moment of inertia about the long cylindrical axis are concerned.
Billy: Wait, that means the equation for the moment of inertia of a solid disk
is the same as for a solid cylinder.
Bo: And the equation for the moment of inertia of a thin, hollow cylinder is the same as for a thin ring.
Bobby: The length is not in those moment of inertia equations.
Billy: It kind of makes sense. The length of the cylinders does not affect where the pieces of the rigid objects are
relative to the axis of rotation.
Booby: Yeah.
Mr. p: I agree.
Let's move on to the moment of inertia of a solid sphere about its center of mass, a lacrosse ball for example.
Bo, how do you think the fraction multiplied by mass times radius squared for the
moment of inertia of a solid sphere about its center of mass compares to the fraction for a solid cylinder?
Bo: Well, because the sphere is solid,
then a larger percentage of its mass is located closer to its axis of rotation than for a solid cylinder.
So the fraction for the equation for moment of inertia of a solid sphere should be less
than the fraction for a solid cylinder.
Mr.p: The moment of inertia of a solid sphere about its center of mass
equals two fifths times mass times radius squared.
And two fifths is less than one-half. So that is correct.
Billy, what about the last one, a racquetball for example.
Where do you think the fraction for the equation for the moment of inertia of a rigid, thin hollow sphere
of constant density about its center of mass fits in with our other fractions?
Billy: Well, compared to the solid cylinder
a hollow sphere has a larger proportion of its mass located farther from the axis of rotation, so...
Bo: Really. I'm not so sure about that.
Bobby: Actually, I agree with Billy.
A hollow sphere looks like it does have a larger proportion of its mass farther from the axis of rotation.
Bo: I don't think that's true.
Mr.p: Let's assume Billy is correct and see what happens.
Billy: Okay, assuming a hollow sphere has a larger proportion of its mass farther from the axis of rotation,
we would expect the fraction for the equation for the moment of inertia of a
thin, hollow sphere about its center of mass to be more than for a solid cylinder.
However, compared to the thin, hollow cylinder,
a hollow sphere has a smaller proportion of its mass located farther from the axis of rotation, so we would
expect the fraction for the equation for the moment of inertia of a thin, hollow sphere about its center of mass
to be less than for a thin, hollow cylinder.
Mr.p: The moment of inertia of a thin, hollow sphere about its center of mass equals
two-thirds mass times radius squared.
Two thirds is greater than one half and less than one, so, Billy, you are correct.
We have now completed discussing all six moments of inertia equations,
and I'll just take a moment to point out an error students often make,
which is to forget that the thin rod moment of inertia equations include L for length and
all the others include R for radius instead.
It is not unusual for students to think that the thin rod equations also have R in them,
but they don't, they have L for length.
But again, please do not memorize these equations.
Instead, understand why they have their relative fractions in their moment of inertia equations.
Thank you very much for learning with me today, I enjoyed learning with you.