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--- 183_notes:rot_ke [2022/10/31 14:22] – valen176
+++ 183_notes:rot_ke [2023/11/07 16:03] – hallstein
@@ Line 26: / Line 26: @@
 ==== The Rotation of Rigid Objects ====
-{{ 183_notes:week10_rotational1.png?400}}
+{{ 183_notes:week10_rotational1a.png?400}}
 The merry-go-round is an example that demonstrates that you need to keep track of how far objects are from the center of mass when they are rotating. But to determine the kinetic energy of the merry-go-round can be tough because we have to consider how each atom contributes to the kinetic energy. You will read how to do that in a bit, but for now consider the system in the figure to the right that rotates at a constant angular speed, $\omega$.
@@ Line 105: / Line 105: @@
 | A hollow spherical shell with mass $m$ and radius $R$ spun around any axis| {{183_notes:moment_of_inertia_hollow_sphere.svg.png?200}} | $I = \dfrac{2}{3} mR^2$ |
-==== Example ====
+==== Examples ====
 [{{ 183_notes:semi_hollow_sphere.png?220|A Semi-Hollow Sphere}}]
-Suppose that you wanted to find the moment of inertia of a semi-hollow sphere (shown to the right) with outer radius R, inner radius r, and uniform density d rotating about its center. There is no obvious equation for this but you do have an equation for the moment of inertia of a solid sphere, which is $I = \dfrac{2}{5} mR^2$. You can find the moment of inertia of the whole sphere *as if it were not hollow* ($I_R$) then subtract the moment of inertia of the inner sphere that is really just empty space ($I_r$), that is:
+Suppose that you want to find the moment of inertia of a semi-hollow sphere (shown to the right) with outer radius R, inner radius r, and uniform density d rotating about its center. There is no obvious equation for this but you do have an equation for the moment of inertia of a solid sphere, which is $I = \dfrac{2}{5} mR^2$. To find the moment of inertia of the semi-hollow sphere ($I_{\text{semi-hollow}}$) you can find the moment of inertia of the outer sphere as if were is **not** hollow ($I_R$), then subtract the moment of inertia of the inner sphere ($I_r$) as if it is not really empty space, that is:
-$$I_{\text{semi-hollow}} = I_R - I_r$$
+$$I_{\text{semi-hollow}} = I_R - I_r = \frac{2}{5}m_R R^2 - \frac{2}{5}m_r r^2$$
-To find $I_R$ and $I_r$, you can use the density
+However, you don't know the masses of the outer and inner spheres ($m_R$ and $m_r$), so you will need to find them. This can be done by first finding their volumes ($V_R$ and $V_r$) with the volume of a sphere equation:
-Other examples:
+$$ V_R = \frac{4}{3}\pi R^3, V_r = \frac{4}{3}\pi r^3$$
+Then multiplying these volumes by the density of the sphere material:
+$$ m_R = \frac{4}{3}\pi R^3d, m_r = \frac{4}{3}\pi r^3d $$
+Plugging these back into the moment of inertia equation gives:
+$$ I_{\text{semi-hollow}}  = \frac{2}{5}(\frac{4}{3}\pi R^3d)R^2 - \frac{2}{5}(\frac{4}{3}\pi r^3d)r^2 = \frac{8}{15}\pi d(R^5 - r^5)$$
+More examples:
   * [[:183_notes:examples:The Moment of Inertia of a Diatomic Molecule]]
   * [[:183_notes:examples:The Moment of Inertia of a Bicycle Wheel]]