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| 183_notes:center_of_mass [2014/09/25 14:20] – [Motion of the center of mass] caballero | 183_notes:center_of_mass [2022/12/22 15:43] (current) – 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 " |
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| + | ==== Lecture Video ==== | ||
| + | {{youtube> | ||
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| + | ==== The Center of Mass ==== | ||
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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 ==== | ||
| - | 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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| + | 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. |
| $$\vec{r}_{cm} = \dfrac{m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3}{m_1+m_2+m_3} = \dfrac{1}{M_{tot}} \left(m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3\right)$$ | $$\vec{r}_{cm} = \dfrac{m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3}{m_1+m_2+m_3} = \dfrac{1}{M_{tot}} \left(m_1 \vec{r}_1+m_2 \vec{r}_2+m_3 \vec{r}_3\right)$$ | ||
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| $$\vec{r}_{cm} = \dfrac{\sum_i m_i \vec{r}_i}{\sum_i m_i} = \dfrac{1}{M_{tot}} \left(\sum_i m_i \vec{r}_i\right)$$ | $$\vec{r}_{cm} = \dfrac{\sum_i m_i \vec{r}_i}{\sum_i m_i} = \dfrac{1}{M_{tot}} \left(\sum_i m_i \vec{r}_i\right)$$ | ||
| - | [{{ 183_notes: | + | [{{ 183_notes: |
| The center of mass is a construct, there might be an object located at the center of mass, but there doesn' | The center of mass is a construct, there might be an object located at the center of mass, but there doesn' | ||
| - | The horseshoe example leads to a different way of calculating the center of mass. Albeit, you can still think about it as adding all the bits of mass weighted by their locations. In the case of the horseshoe the system is a bunch of atoms that make up the horseshoe. But we don't add them up in discrete pieces ((We could, but we'd only get an approximate answer where the accuracy depends on how small the chunks are)). Instead, we take infinitesimally small chunks of mass ($dm$) and sum over them continuously (using an integral). | + | The horseshoe example leads to a different way of calculating the center of mass. Albeit, you can still think about it as adding all the bits of mass weighted by their locations. In the case of the horseshoe the system is a bunch of atoms that make up the horseshoe. But we don't add them up in discrete pieces ((We could, but we'd only get an approximate answer where the accuracy depends on how small the chunks are.)). Instead, we take infinitesimally small chunks of mass ($dm$) and sum over them continuously (using an integral). |
| $$r_{cm} = \dfrac{\int \vec{r} dm}{\int dm} = \dfrac{1}{M_{tot}}\left(\int \vec{r} dm\right)$$ | $$r_{cm} = \dfrac{\int \vec{r} dm}{\int dm} = \dfrac{1}{M_{tot}}\left(\int \vec{r} dm\right)$$ | ||
| - | This form can be challenging to use because will have to determine $dm$ in terms of the position vector and decide what the limits of the integral will be. There' | + | This form can be challenging to use because |
| ==== Motion of the center of mass ==== | ==== Motion of the center of mass ==== | ||
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| $$M_{tot}\vec{v}_{cm} = \sum_i \vec{p}_i = \vec{p}_{sys}$$ | $$M_{tot}\vec{v}_{cm} = \sum_i \vec{p}_i = \vec{p}_{sys}$$ | ||
| - | So you can think of the velocity of the center of mass ($\vec{v}_{cm}$) as a single particle that has the total mass of the system ($M_{tot}$), | + | So you can think of the velocity of the center of mass ($\vec{v}_{cm}$) as a single particle that has the total mass of the system ($M_{tot}$), |
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| + | === Simulation of the motion of the center of mass === | ||
| + | The simulation below shows a binary star (red star and yellow star) system where the total momentum of the system is non-zero, but because there are no external forces to the two particle system, the center of mass moves with constant momentum (green | ||
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| + | /* | ||
| + | {{url> | ||
| + | */ | ||
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| + | {{youtube> | ||
| + | ==== Examples ===== | ||
| + | * [[: | ||