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How to Calculate the Moment of Inertia of a Rigid Body?

Answer #1

Level 3 (with higher mathematics)
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So you want to calculate the moment of inertia \( I \) of a rigid system rotating around a fixed axis. Of course, you have to define the axis of rotation. The moment of inertia counteracts the rotational motion and depends on the mass distribution of the system, i.e. on the mass distributed around the axis of rotation. Here we have to distinguish if the system has a discrete mass distribution (e.g. planets orbiting the sun) or if it has a continuous mass distribution (e.g. a rotating cylinder).

In a discrete mass distribution, you have \(N\) masses distributed in space and having different distances \(r\) from the axis of rotation. To determine the moment of inertia \( I \) for this discrete rotating system, you need to know the individual masses \( m_i \) and their distances perpendicular to the axis of rotation \( r_{\perp i} \):

Moment of inertia of a discrete mass distribution
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In a continuous mass distribution, you don't have discrete masses, but the total mass is "smeared" out over a certain region of space. For example, a hollow cylinder has a continuous mass distribution where its mass is smeared out on its surface. To calculate the moment of inertia \( I \) of such a system, you have to replace discrete summation in 1 by an integral and the masses \(m_i\) by the mass density \(\rho(\boldsymbol{r})\). Then you integrate over the volume \(V\) of the rotating system:

Moment of inertia of a continuous mass distribution
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The given mass density \(\rho(\boldsymbol{r})\) for a three-dimensional system depends on the position vector \(\boldsymbol{r} \) and the volume element \(\text{d}v\), in which the mass \(\rho(\boldsymbol{r}) \text{d}v\) is located, has the perpendicular distance \( r_{\perp} \) to the axis of rotation.

If the theoretical determination using the integral is difficult (because the system has no symmetries), then the moment of inertia can be determined experimentally. For this purpose, the total torque \(M\) and the angular acceleration \(\alpha\) are measured and the moment of inertia \(I\) is calculated using their ratio:

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Torque and Angular Acceleration of a Disk
Torque and angular acceleration of a disk (continuous mass distribution).