Calculus

Applications of Integrals

Applications of Integrals Logo

Moment of Inertia

The moment of inertia (also called the second moment) is a physical quantity which measures the rotational inertia of an object.

The moment of inertia can be thought as the rotational analogue of mass in the linear motion.

The moment of inertia of a body is always defined about a rotation axis.

Moment of Inertia of Point Masses

For a single mass, the moment of inertia is expressed as

\[I = m{r^2},\]

where \(m\) is the mass of the object, and \(r\) is the distance from the object to the axis of rotation.

If a system consists of \(n\) bodies, then the moment of inertia is given by

\[I = {m_1}r_1^2 + {m_2}r_2^2 + \cdots + {m_n}r_n^2 = \sum\limits_{i = 1}^n {{m_i}r_i^2} ,\]

where \({m_1},{m_2},\ldots,{m_n}\) are the masses of the bodies, \({r_1},{r_2},\ldots,{r_n}\) are their distances from the axis of rotation.

We can represent the last equation in the form

\[I = MR_g^2,\]

where \(M = \sum\limits_{i = 1}^n {{m_i}} ,\) and \({R_g}\) is called the radius of gyration.

It follows from the above equation that

\[{R_g} = \sqrt {\frac{I}{M}} .\]

When all masses \({m_i}\) are the same:

\[{m_1} = {m_2} = \cdots = {m_n} = m,\]

then the moment of inertia can be written in the form

\[I = m\left( {r_1^2 + r_2^2 + \cdots + r_n^2} \right) = m\sum\limits_{i = 1}^n {r_i^2} .\]

In this case,

\[{R_g} = \frac{1}{n}\sqrt {r_1^2 + r_2^2 + \cdots + r_n^2} = \frac{1}{n}\sqrt {\sum\limits_{i = 1}^n {r_i^2} } ,\]

where \(n\) is the number of bodies in the system.

Moment of Inertia of a Lamina

When we deal with distributed objects like a lamina, or a solid, we need to calculate the contribution of each infinitesimally small piece of mass \(dm\) to the total moment of inertia \(I.\) This can be done through integration. In general case, finding the moment of inertia requires double integration or triple integration. However, in some special cases, the problem can be solved using single integrals.

Case 1. Density Depends on the \(x-\)Coordinate

Let a planar lamina be bounded by the curves \(y = f\left( x \right)\) and \(y = g\left( x \right)\) on the interval \(\left[ {a,b} \right].\) Suppose that the lamina is rotated about the \(y-\)axis.

Moment of inertia of a lamina about the y-axis with the density depending on the x-coordinate.
Figure 1.

If the density \(\rho\) only depends on the \(x-\)coordinate, then the moment of inertia of a thin rectangle of width \(dx\) is defined by the formula

\[dI = {x^2}dm = {x^2}\rho \left( x \right)\left[ {f\left( x \right) - g\left( x \right)} \right]dx.\]

The total moment of inertia of the lamina about the \(y-\)axis is given by the integral

\[I = \int\limits_a^b {{x^2}\rho \left( x \right)\left[ {f\left( x \right) - g\left( x \right)} \right]dx} .\]

Case 2. Density Depends on the \(y-\)Coordinate

Similarly, we can consider a region of type \(II,\) bounded by the curves \(x = f\left( y \right),\) \(x = g\left( y \right)\) and the horizontal lines \(y = c,\) \(y = d.\) If the density of such a region only depends on the variable \(y,\) that is \(\rho = \rho \left( y \right),\) then the moment of inertia \(I\) of the lamina can be expressed by the single integral

\[I = \int\limits_c^d {{y^2}\rho \left( y \right)\left[ {f\left( y \right) - g\left( y \right)} \right]dy} .\]
Moment of inertia of a lamina about the x-axis with the density depending on the y-coordinate.
Figure 2.

Parallel Axis Theorem

Suppose that an object is rotated about an axis passing through the center of gravity of the object and has the moment of inertia \({I_C}.\) Then the moment of inertia \(I\) about any other axis of rotation, which is parallel to the initial axis is given by the parallel axis theorem (also known as Huygens–Steiner theorem):

\[I = {I_C} + m{d^2},\]

where \(m\) is the mass of the object, and \(d\) is the distance between the two axes.

Parallel Axis Theorem
Figure 3.

By definition, the distance \(d\) is the perpendicular distance between the axes.

Solved Problems

Click or tap a problem to see the solution.

Example 1

Find the moment of inertia of a rectangle with sides \(a\) and \(b\) with respect to an axis passing through the side \(b.\)

Example 2

Find the moment of inertia of the semi-circular arc of radius \(R\) and mass \(m\) about an axis passing through its diameter.

Example 3

Find the moment of inertia of a uniform thin disk of radius \(R\) and mass \(m\) rotating about an axis passing through its center.

Example 4

A thin uniform rod of length \(\ell\) and mass \(m\) is rotated about the axis which is perpendicular to the rod and passes through its end. Calculate the moment of inertia of the rod.

Example 1.

Find the moment of inertia of a rectangle with sides \(a\) and \(b\) with respect to an axis passing through the side \(b.\)

Solution.

Moment of inertia of a rectangle with sides a and b.
Figure 4.

Consider a small strip of the rectangle of width \(dx.\) The distance of the strip from the axis of rotation is equal to \(x.\) Therefore, it has the moment of inertia

\[dI = {x^2}dm = {x^2}\rho bdx.\]

Assuming the density is \(\rho = 1,\) we can write

\[dI = b{x^2}dx.\]

Integrating from \(x = 0\) to \(x = a\) yields:

\[I = \int\limits_0^a {b{x^2}dx} = b\int\limits_0^a {{x^2}dx} = \left. {\frac{{b{x^3}}}{3}} \right|_0^a = \frac{{b{a^3}}}{3}.\]

Example 2.

Find the moment of inertia of the semi-circular arc of radius \(R\) and mass \(m\) about an axis passing through its diameter.

Solution.

Moment of inertia of the semi-circular arc of radius R about the diameter.
Figure 5.

We take the radius vector that forms an angle \(\theta\) with the positive direction of the \(x-\)axis and consider an infinitely small element \(d\ell\) of the arc which is determined by increment \(d\theta.\) The mass of the element \(d\ell\) is

\[dm = \frac{{d\ell}}{\ell}m = \frac{m}{{\pi R}}d\ell.\]

The moment of inertia of the element \(dm\) of the arc about the \(y-\)axis is given by

\[dI = {x^2}dm = \frac{{m{R^\cancel{2}}{{\cos }^2}\theta }}{{\pi \cancel{R}}}d\ell = \frac{{mR\,{{\cos }^2}\theta }}{\pi }d\ell.\]

Recall that \(d\ell = Rd\theta.\) Then

\[dI = \frac{{m{R^2}{{\cos }^2}\theta }}{\pi }d\theta .\]

To calculate the total moment of inertia of the semi-circular arc, we integrate from \(\theta = - \frac{\pi }{2}\) to \(\theta = \frac{\pi }{2}:\)

\[I = \int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\frac{{m{R^2}{{\cos }^2}\theta }}{\pi }d\theta } = \frac{{m{R^2}}}{\pi }\int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {{{\cos }^2}\theta d\theta } = \frac{{m{R^2}}}{{2\pi }}\int\limits_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\left( {1 + \cos 2\theta } \right)d\theta } = \frac{{m{R^2}}}{{2\pi }}\left. {\left( {\theta + \frac{{\sin 2\theta }}{2}} \right)} \right|_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} = \frac{{m{R^2}}}{{2\pi }}\left[ {\frac{\pi }{2} - \left( { - \frac{\pi }{2}} \right)} \right] = \frac{{m{R^2}}}{2}.\]

Example 3.

Find the moment of inertia of a uniform thin disk of radius \(R\) and mass \(m\) rotating about an axis passing through its center.

Solution.

Calculating the moment of inertia of a thin uniform disk of radius R and mass m.
Figure 6.

Take an arbitrary thin ring of radius \(0 \lt r \lt R\) and thickness \(dr.\) The mass of the elementary ring is

\[dm = \rho dA = 2\pi\rho rdr,\]

where \(\rho\) is the density of the disk material, and \(dA\) is the area of the ring.

The moment of inertia of the ring is given by

\[dI = {r^2}dm = 2\pi \rho {r^3}dr.\]

To find the moment of inertia of the entire disk, we integrate from \(r = 0\) to \(r = R:\)

\[I = 2\pi \rho \int\limits_0^R {{r^3}dr} = 2\pi \rho \left. {\frac{{{r^4}}}{4}} \right|_0^R = \frac{{\pi \rho {R^4}}}{2}.\]

Note that the mass of the disk is

\[m = \rho A = \pi \rho {R^2},\]

so

\[I = \pi \rho {R^2} \times \frac{{{R^2}}}{2} = \frac{{m{R^2}}}{2}.\]

Example 4.

A thin uniform rod of length \(\ell\) and mass \(m\) is rotated about the axis which is perpendicular to the rod and passes through its end. Calculate the moment of inertia of the rod.

Solution.

First we determine the moment of inertia of the rod about the axis passing through the center of gravity.

Moment of inertia of a thin rod about the axis passing through the center of gravity.
Figure 7.

Consider a small portion of the rod located at distance \(x\) from the center. If the width of the element is \(dx,\) then the moment of inertia of the element about the center is

\[d{I_C} = \rho {x^2}dx,\]

where \(\rho\) is the linear density of the rod.

The total moment of inertia is defined through integration:

\[{I_C} = \int\limits_{ - \frac{l}{2}}^{\frac{l}{2}} {\rho {x^2}dx} = \left. {\frac{{\rho {x^3}}}{3}} \right|_{ - \frac{l}{2}}^{\frac{l}{2}} = \frac{\rho }{3}\left[ {\frac{{{l^3}}}{8} - \left( { - \frac{{{l^3}}}{8}} \right)} \right] = \frac{{\rho {l^3}}}{{12}}.\]

Since the mass of the rod is

\[m = \rho l,\]

we have

\[{I_C} = \frac{{\rho {l^3}}}{{12}} = \rho l \times \frac{{{l^2}}}{{12}} = \frac{{m{l^2}}}{{12}}.\]

Suppose now that the rod is rotated about the axis passing through one of the ends.

Moment of inertia of a thin rod about the axis passing through its end.
Figure 8.

The new axis is located at a distance \(d = \frac{l}{2}\) from the center of the rod. Using the parallel axis theorem, we get

\[I = {I_C} + m{d^2} = \frac{{m{l^2}}}{{12}} + m{\left( {\frac{l}{2}} \right)^2} = \frac{{m{l^2}}}{{12}} + \frac{{m{l^2}}}{4} = \frac{{m{l^2}}}{3}.\]

See more problems on Page 2.

Page 1 Page 2