What is moment of inertia?
The moment of inertia might be a difficult concept for students, but with the right analogies it can be understood easily. In essence, the moment of inertia is the rotational analogy for the mass of an object.
Newton’s Second Law
The mass of an object is an important quantity for several equations. The most famous one is Newton’s second law: F = m*a.
But what if we have a rotating body? In that case we have an analogous formula, which is Newton’s second law for rotation: τ = I*α, where τ is the torque (analogous to the force) and α is the angular acceleration (analogous to the linear acceleration). Therefore, I is a kind of “rotational mass”, which we call moment of inertia.
Momentum
The linear momentum is defined as p=m*v. The momentum of a system remains constant if there are no forces applied. Therefore, the momentum is a useful formula that can be used to calculate the mass or velocity of the objects that make up a system.
But in the case that the objects are rotating, we need another formula, which is the angular momentum: L=I*ω, where ω is the angular velocity. Again, we see here that the moment of inertia I is the rotational analogy of the mass m.
Kinetic energy
The linear kinetic energy is defined by KE=1/2*m*v2. The total energy of a system (kinetic + potential) is normally conserved, so KE is useful in many exercises.
If the body is rotating, then there is additional energy going into making the rotation possible, and we account by the rotational kinetic energy: KE=1/2*I*ω2. Once again, we see the analogy of mass m and moment of inertia I.
To better understand the analogy: If an object has a large mass, like a car, it is more difficult to move than an object with a lesser mass. In the same sense, an object with a larger moment of inertia (rotational mass) is more difficult to rotate.
If we imagine a marble tied to a string, we can rotate it easily. However, if we tie a football with a rope, we can see that it is much more difficult to rotate, since it has more mass. So, shouldn’t the moment of inertia simply be the mass of the object, since it behaves the same way? The answer is no, because there is another property that influences the moment of inertia: the distribution of the mass.
What do we mean by distribution of the mass? By that we mean how far the mass is from the axis of rotation. So, if we take the same football and tie it to a rope that is double the length, then the football will be double the distance from us, and it will be much harder to rotate.
This means that for the moment of inertia, both the mass and the distance to the mass (or how the mass is distributed with respect to the axis of rotation) is important. In other words, the mass, the shape and the axis of rotation influence the moment of inertia of a rotating body.
Moment of Inertia formula
Definition: The moment of inertia is a quantity that determines the torque that is needed to achieve a certain angular acceleration about a given axis. This is analogous to how the mass determines the force needed for a certain linear acceleration. The moment of inertia depends on the mass distribution and on the axis of rotation.
How To Calculate Moment of Inertia
Step 1: Express dm as a function of r with the help of the density.
Step 2: Find the integral limits with respect to the axis of rotation.
Step 3: Integrate.
Step 4: Substitute the density to get the answer in terms of the mass of the body.
Examples
Example 1
A rod of length L and mass m is rotated about its centre as seen in the picture. Calculate its moment of inertia.
Step 1: The density is defined as ρ = m/V, so in terms of mass we have: m = ρ*V. Now, because in our example we have a rod, we have a one-dimensional object, so instead of Volume we use Length. And instead of density ρ, which is defined for three-dimensional objects, we use λ, which is linear density.
We therefore have the following expression for the mass: m = λ*L.
But we want to have the expression for an infinitesimal mass dm. Because the linear density is constant, what we get is simply: dm = = λ*dL.
Note that in the integral our variable is r, so we can set dL = dr, because it doesn’t matter how we call our distance. What matters is that we use the same letter so that we can integrate with respect to it. So, in the end we have dm = = λ*dr.
Step 2: For the integral limits, we have to count with respect to the axis of rotation. As seen in the picture, the axis is in the middle of the rod, so the lower limit will be -L/2 and the upper limit will be L/2.
Step 3: Integration:
Step 4: We now want to substitute the density to get an answer with the mass of the rod. Because λ=m/L, we get:
Which is the answer.
Example 2
A cylinder of length L, radius R and mass m is rotated about its axis as seen in the picture. Calculate its moment of inertia.
Step 1: Because the cylinder is a three-dimensional body, we can use the following definition of density: ρ = m/V, so in terms of mass we have: m = ρ*V. But we want dm, so we write: dm = ρ*dV.
We now have to find dV. We do so by imagining a “shell”, as seen in the image. The are of that shell is the circumference times dr, in other words 2 π r * dr. To get the volume, what we do is to multiply by L: V = 2 π r L * dr, so in the end we have for the infinitesimal mass: dm = 2 ρ π r L *dr.
Step 2: The integral limits in this case are r = 0 and r = R, because we start at the axis and go to the radius R.
Step 3: Integration:
Step 4: We now want to substitute the density to get an answer with the mass of the rod. Because ρ = m/V and V = π R2 L, we get for the moment of inertia:
Which is the answer.
Things to keep in mind
- The units of the moment of inertia are [kg*m2]. Therefore, if your result doesn’t have those units, then you know you have made a mistake.
- The integration limits depend on how you set up your dm. That comes down to setting up your dV, and there are several ways to calculate the volume of an object. However, you should use the one where you have a dr, because in the integral we want to integrate with respect to r.
- You might be used to setting the limits of integration from 0 to R, but as in Example 1, that is not always the case. So, make sure the limits are correct.
Helpful resources
A good way to check your results is with an online tool. This Online Moment of Inertia Calculator helps you calculate it for different shapes and dimensions and get a numerical result.
If you don’t want a numerical result but simply want to check the formulas for different bodies, this website has a large list of shapes with the moment of inertia with respect to different axis.
Conclusion
The moment of inertia is a very important property of rotating bodies. It is the analogy for mass when we have linear motion. Some applications of the moment of inertia are:
-The beams that are often used for construction are called H-beams. Their shape makes the moment of inertia very large, so that it is very difficult to bend them, thus being able to support heavy objects above them.
– An automobile with a flywheel, which is a large mass fixed on an engine’s crankshaft. The flywheel’s moment of inertia is extremely high, which aids in energy storage.
– Shipbuilding: It could be that a ship sinks by rolling, but it will never sink by pitching. The reason is, the moment of inertia over the pitching axis is much higher compared to that over the rolling axis, so it would take too much effort to rotate it around that axis.
