Let’s talk about mass moment of inertia. It’s a measure of an object’s resistance
to angular acceleration. We’re going to look at a few examples to see how we can find the mass
moment of inertia. With the first few examples, we will cover how to find the mass moment of inertia
for random objects, and the last few will cover composite objects and using the parallel axis
theorem. Imagine we have an object like this. We can find the mass moment of inertia about the zed
axis using this equation. R is the perpendicular distance from the zed axis to a random mass
element. If we add up all of these tiny mass elements, that makes up our whole object. Now
suppose we have to consider the density of this object, what can we do? If you think back to
math class, you might remember that the mass of an object is equal to the density of the object
multiplied by the volume. We represent density with the Greek letter rho. So what is the volume
of our mass element? Well that’s the density times the volume of our element. In simple terms, the
mass of this tiny piece is equal to the density times the volume of this tiny piece. In this case,
the mass element is now equal to rho dv. Keep this in mind because this is our starting step
for all our problems. So we can replace dm with this equation whenever we consider density. Then
our equation becomes this. So as before, adding up all the elements makes up our object, but now,
we consider the density as well. This adding up all the pieces is what the integral does. Most
of the time, objects will have a constant density, so when you do the integral, you can pull it out.
Now, there are 2 ways of calculating mass moment of inertia. Instead of using random weird looking
tiny pieces, we can either use a shell or a disk. That way, we know the volume equation
for either of the 2 objects. So let’s think about the shell method. What we do is place
a bunch of shells inside an object and add it all up to get an answer. In this case,
the volume of the shell element is the following. Zed is the height, and y is the radius. The next
method is using a disk element. In this case, we place a bunch of disks and add it all up.
The volume for it is the following. Here, y is the radius. This method is a bit more tedious since we
first have to determine the mass moment of inertia for the element and then integrate that value.
Now what we talked about works when the axis we are looking at, so x, y, and zed axes goes
through the center of mass of our object. If it doesn’t, we need to use something called
the parallel axis theorem. So imagine we have an object like this, and we are finding the moment of
inertia about this axis. Notice that it doesn’t go through the center of mass. In that case, we
can use this equation to get our answer. The first part is the mass moment of inertia of the object
about the axis that goes through the center of mass. The second part is the mass of the object,
times the distance between the parallel axes. So we just tag on this extra part and we can
find the mass moment of inertia about an axis that doesn’t pass through the center of mass.
Lastly, you should keep in mind these 3 equations. The first tells us how to find the mass moment of
inertia for a rectangular plate, the second for a slender rod, and the 3rd for a circular
disk. With these 3 equations, we can find the mass moment of inertia for composite bodies.
All of this will make more sense with examples, so let’s get started.
Here, we have a cylinder and we need to find the mass moment of inertia about the zed axis.
Looking at the shape, the easiest method to use is the shell method. By following a set of steps,
we can get to our answer. First, we need to write the mass element for our cylinder. Then we replace
the volume element with the volume element of our shell. Then we use our mass moment of inertia
equation. Lastly, we write our answer in terms of mass. Following these steps will lead to an answer
whenever we need to find the mass moment of inertia. So first, what’s the mass element of
our shell? Well, that’s rho, so density, times the volume element. Now, since we aren’t using
random looking pieces and we are using a shell, we know the volume of a shell element. So we’re
going to replace dV with that value. Looking at our diagram, we see that the height is actually h,
and y can be replaced by r, which is the radius, so let’s do that. Don’t forget to change dy to
dr when you do this. Now we can write down our equation for the mass moment of inertia.
Notice how we can replace dm with the value we found before. So let’s do that. The lower bound
starts at 0, and the upper bound is the radius of our cylinder. Now, we bring all the constants out,
so that includes the density, pi, and the height. Let’s do the integral and solve. Now we simplify.
So what we have is the mass moment of inertia, but we don’t want to leave it like this. Instead,
we want to write it in terms of mass. To do that, we need to figure out the mass of our cylinder.
So we go back to the equation we had before and integrate it. So remember, we are just adding up
all the tiny mass elements, in this case, shells, to get the mass of our whole object. Let’s solve
and simplify. This is the mass of our cylinder. Now how do we write our mass moment of inertia in
terms of mass? To do so, we need to divide the mass moment of inertia by the m value.
Imagine you have a value of 45, and m is equal to 5. If we divide 45 by 5,
we get 9. So we can say 45 is equal to 9m because m is equal to 5. That’s exactly what we are doing,
we are writing our mass moment of inertia in terms of m. So let’s divide.
So now, we can write our mass moment of inertia in terms of mass. if you ever want to go back,
you just replace m with our previous value and we get the same result as before. This
tells us the mass moment of inertia for a cylinder about the zed axis. So now if you
ever want to figure out a value, all you need is the mass of the cylinder and it’s radius,
then you plug them into the equation and solve to get the mass moment of inertia about the zed axis.
The result we got is very important when we use the disk method in the next few examples.
Let’s look at this problem where we have a cone and we need to find the mass moment of inertia
about the x axis. For this problem, we will use the disk method. Let’s draw a different
perspective to our diagram so we can see our shape and disk better. So as before, we
will follow our set of steps. First, what is our mass element for the shell? That’s rho dv. Next,
we need to replace the volume element with the volume of a disk because we aren’t using random
looking objects, rather a nice-looking disk and we know the volume of a disk. So in simple terms,
if we integrate it, or add up all the disks, we get the mass of our cone. We can replace y
with the equation given to us. Let’s simplify. Unlike the case when we used a shell element,
we need to take extra steps with the disk method. Remember how in the previous question, we found
the moment of inertia for a cylinder? Well, we have to use that result to find the moment of
inertia for the disk element. I know this can be confusing, but when we do another example,
it should give you a good understanding of this. So in simple terms, our disk is now a cylinder,
and we are finding the moment of inertia element for our disk. So we can write our equation like
this. So all we are doing is going one step backwards, so writing everything in terms
of elements. In other words, if you take the integral, we get the mass moment of inertia for
a cylinder. Note that the radius is now actually the y length, so we replaced that as well.
Next, we plug in the dm value we found before since we know the mass. Notice that we have
the y term, and for that, we can plug in the equation given to us. Let’s simplify.
Now that we have the moment of inertia element, we just take the integral to
get our mass moment of inertia. So our lower bound starts at zero, and the upper bound is
h. Let’s take all the constant out. Now we can simply. In simple terms, what we did was to find
the moment of inertia for our disk, and then add all of those up for the entire object to
figure out the moment of inertia for the whole object. This way, we actually don’t need to use
the equation for the mass moment of inertia since we already used it to figure out the mass moment
of inertia for a cylinder, or now, our disk. This is our mass moment of inertia, but as before,
we want it written in terms of m. So to do that, we need to figure out the mass of our cone.
We go back to our equation and take the integral. Let’s simplify. This gives us the mass of our
cone. Now to write our mass moment of inertia in terms of m, we divide it by the value we just got.
So finally, our mass moment of inertia can be written like this,
and that is the mass moment of inertia for this cone about the x axis.
Let’s do another example where we need to find the mass moment of inertia for the sphere about the
x axis. So first, we write down our mass element for our sphere. Then we replace the volume element
with the volume element of a disk. Next, we replace y with the equation of our sphere.
Next, we need to do our extra step since we are using the disk method. So, we need to write down
our mass moment of inertia element for our disk. Now we replace dm with the value we found before.
We also need to replace the y term with our equation. Let’s tidy it up.
Now we can integrate to get our mass moment of inertia for the whole sphere. The lower bound is
-r and upper bound is r, since this is a sphere with a radius ranging from -r to r. Let’s solve
and simplify our integral. Now as before, we need to write our mass moment of inertia in terms of m.
So let’s find the mass of our sphere using our previous equation. Let’s take the integral and
that gives us the mass of our sphere. Lastly, we divide our mass moment of inertia by the mass.
Now we can write it in terms of m. And that is the mass moment of
inertia for a sphere about the x axis. Now let’s move onto another example
where we have to calculate the mass moment of inertia about an axis passing through point A,
perpendicular to our screen. For this, we will use the equations we already know. Now since
this object is made up of 2 slender rods, we need to use the mass moment of inertia equation for
a slender rod. Now since this equation is for the center of the rod, and point A lies at the
edge of the rod, we need to use the parallel axis theorem. Let’s write the equation down.
One thing we need to find before we apply this equation is to find the mass of each
segment. Let’s quickly do that. The rods have a mass of 4 kg per meter,
and the total length of each piece is 0.2 m. Now we can move onto our equation.
Since we have 2 pieces, we will have to sum the mass moment of inertia for each piece.
The distances between the points are 0.1 m and 0.2 m. Remember, our mass moment of inertia
equation for a slender rod is with respect to the center of mass. So we are measure the distance
from center of mass to point A. This section is our mass moment of inertia for a slender rod.
So our length for each member is 0.2 m. The mass is 0.8 kg. Let’s simplify. And
that is the mass moment of inertia for this object about an axis passing through point A.
Let’s take a look at one last example where we need to find the mass moment of inertia of
this object about the y axis. Looking at the diagram, we see that this is an object with
composite bodies, so 2 different shapes combined together. We have the square plates and 2 circles.
So let’s draw the plates with the holes filled in. The center of mass would be here and here. The
distances to the y axis would be 0.2m. Now let’s draw the holes. Each has a radius of 0.1 m, and
of course the centers are right here and here. The distances to the y axis from the center is 0.2 m.
Now for the square plate, our mass moment of inertia equation is this. For the circle piece,
it’s this. So what we need is the mass of each piece. The mass per unit area is given,
so for the square plate, we have the area of the plate times the mass per unit area. Now the
circle piece. The area of a circle is pi r^2. Now what we will do is find the mass moment of
inertia of the square plates, and from that, we are going to subtract the mass moment of
inertia of the circles. Since the y axis is not the center of each piece, we will have to use
the parallel axis theorem. So let’s go through this equation. Notice we have 2 pieces of each,
so 2 square plates and 2 circle holes. Then we have the equation for the square plate,
the length is 0.4. For the parallel axis theorem part, the distances from the center of mass
to the y axis is 0.2 m. Now we subtract the holes, so for that, we have a radius of 0.1 m,
and the distance from the center of mass to the y axis is 0.2 m. Let’s solve.
And that is the mass moment of inertia for this object about the y axis.
That should cover the types of problems you will face when it comes to finding mass moment of
inertia. For simple objects, it’s beneficial to memorize the mass moment of inertia equations,
and for other complex problems, integration will be required. I hope this video helped!
Thanks for watching and best of luck with your studies.