Steiner theorem or parallel axis theorem for calculating the moment of inertia

In the mathematical description of the rotational motion, it is important to know the moment of inertia of the system about the axis. In the general case, the procedure for finding this quantity involves the implementation of the integration process. The so-called Steiner theorem makes it easier to calculate. Consider it in more detail in the article.

What is the moment of inertia?

Before we state the Steiner theorem, we must deal with the very concept of moment of inertia. Suppose there is a certain body of a certain mass and arbitrary shape. This body can be either a material point, or any two-dimensional and three-dimensional object (rod, cylinder, ball, etc.). If the object in question makes a circular motion around a certain axis with constant angular acceleration α, then we can write the following equation:

M = I * α

Here, the quantity M represents the total moment of forces that gives the acceleration α to the entire system. The proportionality coefficient between them - I, is called the moment of inertia. This physical quantity is calculated by the following general formula:

I = ∫ _m (r ² * dm)

Here r is the distance between the element with mass dm and the axis of rotation. This expression means that it is necessary to find the sum of the products of the squares of the distances r ² and the elementary mass dm. That is, the moment of inertia is not a pure characteristic of the body, which distinguishes it from linear inertia. It depends on the distribution of mass over the entire object that rotates, as well as on the distance to the axis and on the orientation of the body relative to it. For example, the rod will have a different I if it is rotated about the center of mass and about the end.

The moment of inertia and Steiner theorem

The famous Swiss mathematician, Jacob Steiner, proved the theorem on parallel axes and moment of inertia, which now bears his last name. This theorem postulates that the moment of inertia for absolutely any rigid body of arbitrary geometry with respect to some axis of rotation is equal to the sum of the moment of inertia about the axis that intersects the center of mass of the body and is parallel to the first, and the product of the mass of the body by the square of the distance between these axes. Mathematically, this formulation is written as follows:

I _Z = I _O + m * l ²

I _Z and I _O are the moments of inertia about the Z axis and the O axis parallel to it, which passes through the center of mass of the body, l is the distance between the lines Z and O.

The theorem allows, knowing the value of I _O , to calculate any other moment I _Z relative to an axis that is parallel to O.

Proof of the theorem

The formula of the Steiner theorem can be easily obtained independently. To do this, consider an arbitrary body on the xy plane. Let the origin go through the center of mass of this body. We calculate the moment of inertia I _O which passes through the coordinate origin perpendicular to the xy plane. Since the distance to any point in the body is expressed by the formula r = √ (x ² + y ² ), then we obtain the integral:

I _O = ∫ _m (r ² * dm) = ∫ _m ((x ² + y ² ) * dm)

Now we will move in parallel the axis along the x axis by a distance l, for example, in the positive direction, then the calculation for the new axis of the moment of inertia will look like this:

I _Z = ∫ _m (((x + l) ² + y ² ) * dm)

We will reveal the full square in brackets and divide the integrands, we get:

I _Z = ∫ _m ((x ² + l ² + 2 * x * l + y ² ) * dm) = ∫ _m ((x ² + y ² ) * dm) + 2 * l * ∫ _m (x * dm ) + l ² * ∫ _m dm

The first of these terms is I _O , the third term, after integration, gives the term l ² * m, but the second term is equal to zero. The zeroing of the indicated integral is due to the fact that it is taken from the product of x by the elements of mass dm, which on average gives zero, since the center of mass is at the origin. As a result, the formula for Steiner’s theorem is obtained.

The considered case on the plane can be generalized to a three-dimensional body.

Verification of the Steiner formula on the example of the rod

We give a simple example on which we demonstrate how to use the theorem considered.

It is known that for a rod of length L and mass m, the moment of inertia I _O (the axis passes through the center of mass) is m * L 2/12, and the moment I _Z (the axis passes through the end of the rod) is m * L 2/3. We verify this data using the Steiner theorem. Since the distance between the two axes is L / 2, then we obtain the moment I _Z :

I _Z = I _O + m * (L / 2) ² = m * L 2/12 + m * L 2/4 = 4 * m * L 2/12 = m * L 2/3

That is, we checked the Steiner formula and got the same value for I _Z as in the source.

Similar calculations can be performed for other bodies (cylinder, ball, disk), while obtaining the necessary moments of inertia, and without integration.

Moment of inertia and perpendicular axis

The theorem considered concerns parallel axes. For completeness, it is also useful to give a theorem for perpendicular axes. It is formulated as follows: for a plane object of arbitrary shape, the moment of inertia relative to the axis perpendicular to it will be equal to the sum of two moments of inertia about two axes mutually perpendicular and lying in the plane of the object, while all three axes must pass through one point. Mathematically, it is written like this:

I _z = I _x + I _y

Here z, x, y are three mutually perpendicular axes of rotation.

A significant difference between this theorem and Steiner's theorem is that it is applicable only to plane (two-dimensional) solid objects. Nevertheless, in practice it is widely used, mentally cutting the body into separate layers, and then adding up the obtained moments of inertia.