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The rotational moment of inertia or, simply, the rotational inertia, is a scalar physical quantity typical of any object that has mass, and that measures how difficult it is to make it rotate around a certain axis of rotation. It is the rotational equivalent of linear inertia and, as such, it is a quantity that expresses the difficulty in changing the speed of an object, whether it is at rest or in motion, with the difference that, in this case, It’s about angular velocity.
This quantity is of great importance in the description of the rotation movement since it allows us to understand the difference in the behavior of bodies that, despite having the same external shape and mass, behave differently when subjected to torque forces. which tend to make them spin. This difference arises from the difference in the distribution of the mass of the body around the axis of rotation. The above implies that the same body can have different moments of rotational inertia depending on its position relative to the axis of rotation, thus giving rise to different formulas to calculate the moment of inertia.
Having said the above, it is clear that there are as many formulas to find the moment of inertia as possible shapes of existing objects and axes of rotation. However, there are some particular cases of regular geometric shapes that rotate around axes that arise naturally in practice. In the following sections, we will see the most important formulas to determine the rotational moment of inertia of these bodies.
Formula for the moment of inertia of a point particle
The moment of inertia of a point particle corresponds to the original definition of this physical quantity. This expression comes from the expression for rotational kinetic energy when it is written in terms of angular velocity, w.
Suppose we have a particle of mass m revolving around a central axis like the following:
The kinetic energy of this particle, like that of any other moving particle, is determined by half the product between its mass and its speed (the magnitude of its speed) raised to the square, that is, 1/2 mv 2 . However, if the only movement that this particle describes is rotation around the axis (there is no translation), we can express the linear speed of the particle as a function of its angular speed, writing v = rω. By doing this, the kinetic energy, which in this case is exclusively rotational kinetic energy, is expressed as:
Where the moment of inertia, I , of the particle is defined as:
In this expression, m is the mass of the point particle and r is the radius of rotation or, what is the same, the distance from the axis of rotation to the particle.
Formula for the moment of inertia of a collection of point particles
Suppose now that we do not have a single particle revolving around an axis, but that we have a system made up of n particles, each one with a particular mass, m i , and each one revolving with a distance r i from the axis of rotation, such as the three-particle system shown below.
If we wanted to calculate the total kinetic energy of this system, we would only have to add the kinetic energies of each of the three particles. If we extend this idea to the general case of n particles and assume that they all move at the same angular velocity (because they rotate together), then the total rotational kinetic energy of the system will be given by:
From where it follows that the total moment of inertia of a system of n particles that revolve together around the same axis, each with its own mass and its own radius of gyration, is given by:
This formula works both for point particles and for spherical particles of any size, as long as the axis of rotation is outside the sphere. If this condition is met, then the radius corresponds to the distance between the axis and the center of the sphere and the mass corresponds to the total mass of the sphere.
Integral formula of moment of inertia of rigid bodies
The above formula for moment of inertia applies to systems formed by point and discrete particles. However, it can be extended to rigid bodies that have a continuous distribution of mass, just as it happens approximately with macroscopic bodies.
In these cases, calculating the moment of inertia consists of dividing the body into small mass elements (Δm i ), each of them located at a distance r i from the axis of rotation, and then applying the previous equation. However, if we push the size of the mass element to the limit where it becomes an infinitesimal element or a mass differential (dm), then the summation becomes the integral, as shown below:
This is the general expression to find the moment of inertia of any rigid body, whatever its shape, or its mass distribution. In most cases, to carry out the integration, the mass element, dm , is replaced by the product of the density of the body multiplied by the volume differential, dV . This allows the integration to be carried out over the entire volume of the rigid body, even if the mass distribution is not uniform (as long as it is known how it varies depending on the position).
In this case, the integral expression of the moment of inertia becomes:
Next, we will present the result of integrating the previous expression for different rigid bodies with regular shapes such as rings, cylinders and spheres, among others. In all the cases described below, the dimensions and masses of the bodies considered are represented with capital letters, in order to distinguish them from the integration variables.
Formula for the moment of inertia of a thin uniform ring of radius R about its central axis
One of the simplest cases when integrating the previous equation is that of a uniform ring that rotates around its center of symmetry. The following figure shows this case.
In the particular case in which the thickness of the ring is negligible compared to its radius, we can consider it as a mass distributed along a circumference without thickness, so that all the mass elements are essentially at the same radius, in In this case, R. Given these conditions, the radius leaves the integral, leaving only the integral of the differential mass, dm, which is simply the mass of the ring, M. The result is:
In this expression, CM indicates that it is the moment of inertia about its center of mass.
Formula for the moment of inertia of a solid sphere of radius R revolving about its center
In the case of a solid sphere of radius R and uniform density, which rotates around any of its diameters (an axis that passes through its center) such as the one shown below, the previous integral can be solved in different ways , among which are using a spherical coordinate system.
The result of the integration in this case is:
Formula for the moment of inertia of a spherical shell of internal radius R 1 and external radius R 2 about its center
If instead of a solid sphere it is a hollow sphere or spherical shell with thick walls, we must consider two radii, the external and the internal. These are shown in the following figure.
In this case, the solution is to consider the spherical shell as a sphere of radius R2 from which a sphere of the same material has been removed from its center whose radius is R1. After determining the mass that the large sphere would have had and that of the small sphere that was withdrawn through the density of the original shell, the inertias of both spheres are subtracted to obtain:
Formula for the moment of inertia of a thin spherical shell of radius R about its center
In the event that the thickness of the spherical shell is negligible compared to its radius or, what is the same, that R 1 is practically equal to R 2 , we can calculate the moment of inertia as if it were a surface distribution of mass, all of it located at a distance R from the center.
In this case we have two options. The first is to solve the integral from scratch. The second is to take the previous result, that of the thick spherical shell, and obtain the limit when R1 tends to R2. The result is as follows:
Formula for the moment of inertia of a thin rod of length L about a perpendicular axis through its center of mass
When we have a thin bar, in essence, we can think of it as a linear distribution of mass, regardless of the shape of its profile (ie, regardless of whether it is a cylindrical, square, or any other shaped bar). In these cases, the only thing that matters is that the dough is distributed evenly along the length of the bar.
In this case, the moment of inertia is expressed as:
Formula for the moment of inertia of a thin rod of length L about a perpendicular axis through one end
This is the same case as above, but with the entire bar rotating around an axis perpendicular from one end:
Since the mass of the bar is, on average, at a greater distance from the axis of rotation, the moment of inertia will be greater. In fact, it is four times greater than the previous case, as shown by the following expression:
Note that in this case the axis does not pass through the center of mass, so the CM subscript of the moment of inertia symbol has been omitted.
Formula for the moment of inertia of a solid cylindrical bar of radius R about its central axis
This case is solved in a very simple way using a cylindrical coordinate system and considering the cylinder as if it were formed by concentric cylindrical shells of equal length, but with different radii. Then the radius is integrated from r = 0 to r = R.
The result of this process is the formula for inertia of a cylindrical bar, which is:
It should be noted that, since this result does not depend on the length of the cylinder, the same expression can be used for the case of a circular disk.
Formula for the moment of inertia of a hollow cylinder of internal radius R 1 and external radius R 2 about its central axis
This case is similar to that of the thick spherical shell. It is applied when the thickness of the shell, or the difference between its external and internal radii is in the same order of magnitude as the radii themselves and, therefore, we cannot consider that the mass is concentrated on a surface. On the contrary, we must consider that it is a three-dimensional distribution of mass along the thickness of the shell.
As in the case of the thick spherical shell, the moment of inertia of a hollow cylinder with an inner radius of R 1 and an outer radius of R 2 can be found by means of direct integration, or by subtracting the moment of inertia from the cylinder that was withdrawn when opening the central hole, of the moment of inertia of a solid cylinder that has the same density as the shell, using the formula of the previous section for each of these two inertias.
The result of either of these two strategies is the same and is presented below:
As in the previous case, since this result does not depend on the length of the cylinder, we can use it to calculate the moment of inertia of a circular disk with a hole in the center, such as, for example, a washer or a Blu-ray disc.
Formula for the moment of inertia of a thin cylindrical shell of radius R about its central axis
In case we have a hollow cylinder like the one shown in the following figure, in which the thickness of the cylindrical shell is very small compared to the radius of the cylinder, we can assume that the mass is distributed only on the surface of radius R .
As in the other cases, we can carry out the direct integration using the areal mass density, or we can evaluate the result of the thick cylindrical shell in the limit where R1 tends to R2. The result is:
Again we note that this result is independent of length. This means that it applies equally to a thin hoop. In fact, we can verify that it is the same result obtained in the section corresponding to a thin ring.
Formula for the moment of inertia of a regular rectangular plate about a perpendicular axis through its center
Finally, consider the case of a rectangular plate that rotates about an axis perpendicular to any of its surfaces, passing through its center of mass, like the one shown below.
The result of direct integration is:
As in the previous cases, this result is independent of the height or thickness of the plate, so it applies equally to a sheet of paper as it does to a solid cement block.
References
Khan Academy. (nd). Rotational Inertia (article) . https://en.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/a/rotational-inertia
OneClass. (2020, October 6). OneClass: Starting with the formula for the moment of inertia of a rod . https://oneclass.com/homework-help/physics/6942744-moment-of-inertia-bar.en.html
Serway, RA, Beichner, RJ, & Jewett, JW (1999). Physics for Scientists and Engineers With Modern Physics: 2: Vol. Volume I (Fifth Edition). McGraw Hill.
Snapsolve. (nd). The moment of inertia of a hollow thick spherical shell . https://www.snapsolve.com/solutions/Themoment-of-inertia-of-a-hollow-thick-spherical-shell-of-mass-M-and-its-inner-r-1681132593667073
