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Hello, I think I've got the right idea on how to perform this question but I just need a little bit of help.
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
[itex]E = \dfrac {1}{2} mv^{2}[/itex]
[itex]v = \omega r[/itex]
[itex] I = \int r^{2} dm[/itex]
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
Homework Statement
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
Homework Equations
[itex]E = \dfrac {1}{2} mv^{2}[/itex]
[itex]v = \omega r[/itex]
[itex] I = \int r^{2} dm[/itex]
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