Hello, I think I've got the right idea on how to perform this question but I just need a little bit of help.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Show that for a rigid body rotating with angular velocity [itex]\omega[/itex] the energy of rotation may be written as:

[itex]E = \dfrac {1}{2}I\omega^{2}[/itex]

where the moment of inertia of the body about the axis of rotation is given by:

[itex]I = \int dV \rho r^{2}[/itex]

where [itex]r[/itex] is the distance from the rotation axis to the volume element [itex]dV[/itex] and [itex]\rho[/itex] is the density of the object in that region

2. Relevant equations

[itex]E = \dfrac {1}{2} mv^{2}[/itex]

[itex]v = \omega r[/itex]

[itex] I = \int r^{2} dm[/itex]

3. The attempt at a solution

I can firstly identify that

[itex]E = \dfrac {1}{2} mv^{2}[/itex] which looks similar to the rotation energy equation.

I know that

[itex]v = \omega r[/itex]

But what's confusing me is that usually, the moment of Inertia is represented as

[itex] I = \int r^{2} dm[/itex] and I don't really know how to link the two.

Substiting angular velocity into the energy equation

[itex] E = \dfrac {1}{2} m(r\omega)^{2}[/itex]

but where do I go now?

Thanks

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Rotational Energy Deriving

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