7.4.1 Intro to Mass Moment of Inertia

Mass moment of inertia, or inertia as it will be referred to from here on, is resistance to rotation. The bigger the inertia, the slower the rotation. [latex]\sum M = I\alpha[/latex]. Inertia is always positive and has units of kgm² or slugft².

For an infinitesimal unit of mass, the inertia depends on how far it is from the axis of rotation.

$$ I = \int_m r^2dm $$

As shown in this image, each little dm at r distance from the axis of rotation (y) is added up (through integration). If r is bigger, the inertia is bigger.

If there is more mass closer to the axis of rotation, the inertia is smaller. A skill that you can develop is your visualization of the rotation about each axis. As shown in the following figure, rotating about the different axes will produce different types of rotation. You can imagine sticking your pencil into an object and twisting along that axis. In this image, rotation about the y axis and x axis produce different types of rotation. Due to the symmetry, rotation about the x axis and z axis looks identical.

The red r’s in this image show the distance that is being measured when adding up each little infinitesimal dm. Notice how the r changees direction from x to y but looks the same between x and z.

Equations have been developed for common shapes so that you don’t have to integrate every time you want to find the inertia of an object. The result is different for each axis, as shown in the following figure.

‘I_xx‘ can be read as ‘the inertia if rotating about the x-axis’. Notice for Ixx and Izz that the height and radius of the cylinder affect the inertia, whereas for Iyy, only the radius is considered.

The equations for each of the objects are listed in a table below. First is a second explanation of inertia.

7.4.2 Inertia Table of Common Shapes

Specific inertia equations depending on the shape of the object and axis of rotation can be found below. Notice some of the shapes have multiple sets of axes: [latex]I_{xx} \text{  and  } I_{xx}^\prime[/latex]. There are multiple equations.

Symmetric Shapes
Thin Ring		$$ I_{xx} = \frac{1}{2}mr^2 \\I_{yy}=mr^2 \\I_{zz} = \frac{1}{2}mr^2  $$ * thickness << 1
Circular Plate		$$ I_{xx} = \frac{1}{4}mr^2 \\I_{yy}=\frac{1}{2}mr^2 \\I_{zz} = \frac{1}{4}mr^2  $$   $$ I_{yy^\prime} = \frac{3}{2}mr^2 $$   * thickness << 1
Cylinder		$$ I_{xx} = \frac{1}{12}m(3r^2+h^2) \\I_{yy}=\frac{1}{2}mr^2 \\I_{zz} = \frac{1}{12}m(3r^2+h^2) $$ $$Volume = \pi r^2 h $$
Sphere		$$I_{xx}=\frac{2}{5}mr^2 \\I_{yy}=\frac{2}{5}mr^2 \\I_{zz}=\frac{2}{5}mr^2 $$   $$Volume = \frac{4}{3}\pi r^3 $$
Slender Rod		$$ I_{xx} = \frac{1}{12}ml^2 \\I_{yy}=0\\I_{zz} = \frac{1}{12}ml^2  $$   $$ I_{xx^\prime} = \frac{1}{3}ml^2 \\I_{zz^\prime} = \frac{1}{3}ml^2  $$ * radius << length
Rectangular Plate		$$ I_{xx} = \frac{1}{12}mh^2 \\I_{yy}=\frac{1}{12}m(h^2+b^2) \\I_{zz} = \frac{1}{12}mb^2  $$ * thickness << 1
Rectangular Block		$$ I_{xx} = \frac{1}{12}m(h^2+d^2) \\I_{yy}=\frac{1}{12}m(d^2+w^2) \\I_{zz} = \frac{1}{12}m(h^2+w^2) $$   $$ Volume = bwh $$
Asymmetric Shapes
Half Cylinder		$$ I_{xx} = \left( \frac{1}{4}-\frac{16}{9 \pi^2} \right)mr^2 + \frac{1}{12}mh^2 \\I_{yy}= \left( \frac{1}{2}-\frac{16}{9 \pi^2} \right) mr^2 \\I_{zz} =\left( \frac{1}{4}-\frac{16}{9 \pi^2} \right)mr^2 + \frac{1}{12}mh^2 $$   $$ I_{xx^\prime} = \frac{1}{12}m(3r^2+h^2) \\I_{yy^\prime} = \frac{1}{2}mr^2)\\I_{zz^\prime} = \frac{1}{12}m(3r^2+h^2) $$   $$Volume = \frac{1}{2} \pi r^2 h $$
Hemisphere		$$I_{xx}=\frac{83}{320}mr^2 \\I_{yy}=\frac{2}{5}mr^2 \\I_{zz}=\frac{83}{320}mr^2 $$   $$I_{xx^\prime}=\frac{2}{5}mr^2 \\I_{zz^\prime}=\frac{2}{5}mr^2 $$   $$Volume = \frac{2}{3}\pi r^3 $$
Cone		$$I_{xx}=\frac{3}{80}m(4r^2+h^2) \\I_{yy}=\frac{3}{10}mr^2 \\I_{zz}=\frac{3}{80}m(4r^2+h^2) $$   $$I_{xx^\prime}=\frac{1}{20}m(3r^2+2h^2) \\I_{zz^\prime}=\frac{1}{20}m(3r^2+2h^2) $$   $$Volume = \frac{1}{3}\pi r^2h $$
Hallow Shells
Cylindrical Shell		$$ I_{xx} = \frac{1}{6}m(3r^2+h^2) \\I_{yy}=mr^2 \\I_{zz} = \frac{1}{6}m(3r^2+h^2) $$   * Thickness << 1
Spherical Shell		$$I_{xx}=\frac{2}{3}mr^2 \\I_{yy}=\frac{2}{3}mr^2 \\I_{zz}=\frac{2}{3}mr^2 $$   * Thickness << 1
Hemispherical Shell		$$I_{xx}=\frac{5}{12}mr^2 \\I_{yy}=\frac{2}{3}mr^2 \\I_{zz}=\frac{5}{12}mr^2 $$   $$I_{xx^\prime}=\frac{2}{3}mr^2 \\I_{zz^\prime}=\frac{2}{3}mr^2 $$   * Thickness << 1
Images source: Jacob Moore et al. http://mechanicsmap.psu.edu/websites/centroidtables/centroids3D/centroids3D.html

Notice how different objects with the same mass and radius rotate at different rates. This simulation shows a cylinder (blue), ring (green), solid sphere (yellow-brown), and spherical shell (red). Which one has the least inertia? Why?

7.4.3 Radius of Gyration

A concept called the radius of gyration (k) converts a shape into a thin ring. This is used for particularly complex shapes. If a homework problem says ‘the radius of gyration k = 15 cm’, that means if the shape were a thin ring, it would have a radius of 15 cm. You calculate the mass moment of inertia using the ring equation: [latex]I = mk^2[/latex]

[latex]\qquad I = mk^2[/latex]

For example, if the mass of an object is m=10 kg, the radius of gyration is 5 m, then the inertia is:

I = mk² = 10 kg * 5 m * 5 m = 250 kgm².

To find the radius of gyration:

$$ k=\sqrt{\frac{I}{m}}=\sqrt{\frac{250 kgm^2}{10kg}} = 5 m $$