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183_notes:center_of_mass [2014/10/31 12:55] – [The center of mass] caballero | 183_notes:center_of_mass [2022/12/21 03:51] – old revision restored (2021/03/04 13:11) hallstein | ||
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+ | Section 3.11 in Matter and Interactions (4th edition) | ||
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===== The Motion of the Center of Mass ===== | ===== The Motion of the Center of Mass ===== | ||
- | In a system of multiple particles, each particle has its own motion. But often, its useful to think about the collective motion of these objects. By " | + | In a system of multiple particles, each particle has its own motion. But often, its useful to think about the collective motion of these objects. By " |
+ | ** | ||
==== Lecture Video ==== | ==== Lecture Video ==== | ||
{{youtube> | {{youtube> | ||
- | ==== The center | + | \\ |
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+ | ==== The Center | ||
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+ | The center of mass is a concept that helps us understand how the motion of a multi-particle system evolves with time. It is connected very strongly to the total momentum of a system as you will read. | ||
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+ | \\ | ||
- | The center of mass is a concept that helps us understand how the motion of a multi-particle system evolves with time. It is connected very strongly to the total momentum of a system as you all read. | ||
=== Flocking birds === | === Flocking birds === | ||
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{{youtube> | {{youtube> | ||
- | It's hard to think about what the motion of a system of objects means without some sort of example. The video above shows the motion of a flock of birds. Each individual bird flies in with it' | + | \\ |
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+ | //(This video is intended for visual learning assistance. Auditory Components are not necessary.)// | ||
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+ | \\ | ||
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+ | It's hard to think about what the motion of a system of objects means without some sort of example. The video above shows the motion of a flock of birds. Each individual bird flies in with its own direction and speed, but the flock (or the bulk) moves in a particular way (it appears to move in a circle or ellipse) that your eye can follow. What you are paying attention to is the motion of the " | ||
=== Calculating the center of mass === | === Calculating the center of mass === | ||
- | The center of mass of a system is the weighted average of the particles in that system. Consider a set of three particles with different mass ($m_i$), which are all located a different locations relative to the origin ($\vec{r}_i$). For these three particles, the center of mass of that system is the vector sum, | + | The center of mass of a system is the weighted average of the particles in that system. Consider a set of three particles with different mass ($m_i$), which are all located a different locations relative to the origin ($\vec{r}_i$). For these three particles, the center of mass of that system is the vector sum. |
$$\vec{r}_{cm} = \dfrac{m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3}{m_1+m_2+m_3} = \dfrac{1}{M_{tot}} \left(m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3\right)$$ | $$\vec{r}_{cm} = \dfrac{m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3}{m_1+m_2+m_3} = \dfrac{1}{M_{tot}} \left(m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3\right)$$ |