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183_notes:rot_ke [2022/10/31 14:20] – valen176 | 183_notes:rot_ke [2023/11/07 16:42] (current) – hallstein | ||
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===== Rotational Kinetic Energy ===== | ===== Rotational Kinetic Energy ===== | ||
- | Earlier, you read about [[183_notes: | + | Earlier, you read about [[183_notes: |
==== Atoms in Rotating Objects Can Move with Different Speeds ==== | ==== Atoms in Rotating Objects Can Move with Different Speeds ==== | ||
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==== The Rotation of Rigid Objects ==== | ==== 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, $\omega$. | 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, $\omega$. | ||
- | The kinetic energy of this system is the sum of individual kinetic energies of the four individual particles. | + | The kinetic energy of this system is the sum of the individual kinetic energies of the four individual particles. |
- | $$K = \dfrac{1}{2}{m_1}\, | + | $$K = \dfrac{1}{2}{m_1}\, |
Each of these velocities can be described in terms of the distance to the particles and the angular speed of the whole apparatus (e.g., $v_1 = r_{\perp 1}\omega$). | Each of these velocities can be described in terms of the distance to the particles and the angular speed of the whole apparatus (e.g., $v_1 = r_{\perp 1}\omega$). | ||
- | $$K = \dfrac{1}{2}{m_1}\, | + | $$K = \dfrac{1}{2}{m_1}\, |
You can group the terms and find that there' | You can group the terms and find that there' | ||
- | $$K = \dfrac{1}{2}\left({m_1}\, | + | $$K = \dfrac{1}{2}\left({m_1}\, |
The use of the symbol $\perp$ here is to remind you that the distance measured from the object to the rotation axis is the perpendicular distance to the axis. | The use of the symbol $\perp$ here is to remind you that the distance measured from the object to the rotation axis is the perpendicular distance to the axis. | ||
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| A hollow spherical shell with mass $m$ and radius $R$ spun around any axis| {{183_notes: | | A hollow spherical shell with mass $m$ and radius $R$ spun around any axis| {{183_notes: | ||
- | ==== Example | + | ==== Examples |
- | + | ||
- | 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 " | + | |
[{{ 183_notes: | [{{ 183_notes: | ||
- | Other examples: | + | 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: |
+ | |||
+ | $$I_{\text{semi-hollow}} = I_R - I_r = \frac{2}{5}m_R R^2 - \frac{2}{5}m_r r^2$$ | ||
+ | |||
+ | 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: | ||
+ | |||
+ | $$ V_R = \frac{4}{3}\pi R^3, V_r = \frac{4}{3}\pi r^3$$ | ||
+ | |||
+ | 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}} | ||
+ | |||
+ | More examples: | ||
* [[: | * [[: | ||
* [[: | * [[: |