Name: calculating rotational inertia in combined systems
Uploaded: 2023-04-30T11:01:25.000Z
Duration: 1 min 7 s
Channel: JoVE

Moment of inertia of an object about an axis is defined as the sum of the products of the mass of each particle in the object with the square of its distance from the axis of rotation.
In case of compound objects, the total moment of inertia of the object is the sum of moment of inertia of individual components along the axis of rotation.
Consider a wheel of radius 0.5 m having a rim of mass 10 kg and twelve spokes each having mass 0.3 kg and length 0.5 m.
Now, to determine the total moment of inertia of the wheel, first, individually determine the moment of inertia of the rim rotating along its cylindrical axis and the spokes rotating about their ends. Then, add the moment of inertia of all the components of the wheel.
By substituting the known values in the equation, the total moment of inertia for the wheel is obtained to be 2.8 kg&#8901;m2.

moment of inertia of compound objects

Moment of Inertia of Compound Objects

The moment of inertia is a quantitative measure of the rotational inertia of an object. It is defined as the sum of the products obtained by multiplying the mass of each particle of matter in a given body by the square of its distance from the axis. The total moment of inertia for compound objects can be found by determining and adding the moment of inertia of individual components together.
Consider a child of mass (mc) 25 kg standing at a distance (rc) of 1 m from the axis of a rotating merry-go-round. The merry-go-round is approximated as a uniform solid disk with a mass (mm) of 500 kg and a radius (rm) of 2 m. Find the moment of inertia of the compound system.
The total moment of inertia of the system can be determined by adding up the individual moments of inertia of the merry-go-round and the child rotating on the axis. Since the mass and size of the child are much smaller than the merry-go-round, the child can be considered as a point mass.
<ol>
	<li>The moment of inertia (I) for the child is calculated as</li>
</ol>
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<ol>
	<li value="2">The moment of inertia (I) for the merry-go-round is calculated as</li>
</ol>
<img alt="Chemical equilibrium reaction, N2 + 3H2 &#8596; 2NH3, formula diagram, illustrating Haber process." src="/files/ftp_upload/12709/12709_Equation_2v2.png" style="text-align: center; height: 38px;" data-altupdatedat="1761830439000" data-alt="Equation2" />
<ol>
	<li value="3">By substituting and adding both values, the total moment of inertia of the system is determined to be 1025 kg&#8901;m2.</li>
</ol>
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia">Openstax, University Physics Volume 1, Section 10.5: Calculating Moments of Inertia.</a>

The moment of inertia is a quantitative measure of the rotational inertia of an object. It is defined as the sum of the products obtained by multiplying ...

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calculating rotational inertia in combined systems

Rotational inertia, also called moment of inertia, describes how hard it is to rotate an object. It depends on how mass is spread out from the axis of rotation. For compound objects, the total moment of inertia is found by adding the moments of inertia of the individual parts.

A simple example uses a child on a rotating merry-go-round. The child has a mass of 25 kg and stands 1 m from the axis. The merry-go-round is modeled as a uniform solid disk with a mass of 500 kg and a radius of 2 m.

To find the moment of inertia of the compound system, each part is handled separately. The child is small compared with the merry-go-round, so the child can be treated as a point mass. The moment of inertia for the child is calculated from its mass and distance from the axis. The moment of inertia for the merry-go-round is calculated from the disk model.

After both values are found, they are added together to get the total moment of inertia. For this system, the result is 1025 kg·m². This example shows how moments of inertia for separate components combine to describe one rotating object.

Calculating Rotational Inertia in Combined Systems

calculating-rotational-inertia-in-combined-systems

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Rotational inertia, also called moment of inertia, describes how hard it is to rotate an object. It depends on how mass is spread out from the axis of rotation. For compound objects, the total moment of inertia is found by adding the moments of inertia of the individual parts. A simple example uses a child on a rotating merry-go-round. The child has a mass of 25 kg and stands 1 m from the axis. The merry-go-round is modeled as a uniform solid disk with a mass of 500 kg and a radius of 2 m. To...

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key terms and concepts

Learn how to calculate the moment of inertia of a compound object by adding the values for each part, using a child and merry-go-round example.

Calculating Rotational Inertia in Combined Systems | JoVE K12

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Calculating Rotational Inertia in Combined Systems

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