Systems and Center of Mass · 系统与质心
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| center of mass/ˈsentə ɒv mæs/ | 质心 | zhì xīn |
| system/ˈsɪstəm/ | 系统 | xì tǒng |
| internal forces/ɪnˈtɜːnl ˈfɔːsɪz/ | 内力 | nèi lì |
| external forces/ekˈstɜːnl ˈfɔːsɪz/ | 外力 | wài lì |
| point particle/pɔɪnt ˈpɑːtɪkl/ | 质点 | zhì diǎn |
| integral/ˈɪntɪɡrəl/ | 积分 | jī fēn |
| net external force/net ekˈstɜːnl fɔːs/ | 净外力 | jìng wài lì |
Throw a spinning wrench
- Toss a wrench across the room and it tumbles wildly -- yet one single point traces a smooth, simple arc.
- That special point is the center of mass 质心.
- No matter how an object spins or wobbles, its center of mass moves as if all the mass were squeezed into that one spot.
- Find it, and a messy tumbling problem becomes a simple point-mass problem.
抛出一把旋转的扳手
- 把扳手横扔过房间,它剧烈翻滚——可是有一个点却划出一条平滑简单的弧线。
- 那个特别的点就是质心。
- 无论物体怎样旋转或摇摆,它的质心运动起来就好像全部质量都被压缩到了那一个点上。
- 找到它,一个乱翻滚的问题就变成了简单的质点问题。
Treat a system as one point
- A system 系统 is any group of objects we choose to study together.
- Forces inside the system are internal forces 内力; forces from outside are external forces 外力.
- The whole system moves as if it were a single point particle 质点 located at its center of mass.
把系统当作一个点
- 系统是我们选择放在一起研究的任意一组物体。
- 系统内部的力是内力;来自外部的力是外力。
- 整个系统运动起来,就好像它是位于质心处的一个质点。
Select all · 所有 true statements about the center of mass. · 选出关于质心的所有正确陈述。
The first two are the core rules. The third is false -- it is the mass · 质量-weighted center, not the shape's midpoint. · 前两条是核心规则。第三条是错误的——它是质量加权中心，而非形状的中点。
Center of mass of discrete masses
- For separate point masses along a line, the center of mass is the mass-weighted average position:
- Heavier masses pull the center of mass toward themselves.
离散质量的质心
- 对于沿一条直线分布的若干质点,质心是按质量加权的平均位置:
- 越重的质量,越把质心拉向自己。
A $1\ \text{kg}$ mass is at $x = 0$ and a $3\ \text{kg}$ mass is at $x = 4\ \text{m}$. Where is the center of mass (in m)? · 质量为$1\ \text{kg}$的物体位于$x = 0$，质量为$3\ \text{kg}$的物体位于$x = 4\ \text{m}$。质心在哪里（单位：m）？
$x_{cm} = \dfrac{1(0) + 3(4)}{1 + 3} = \dfrac{12}{4} = 3\ \text{m}$ -- close to the heavier mass. · $x_{cm} = \dfrac{1(0) + 3(4)}{1 + 3} = \dfrac{12}{4} = 3\ \text{m}$——靠近较重质量。
Move one mass farther from the others. The center of mass... · 将一个质量移远其他质量。质心...
The mass-weighted average follows the mass -- moving a mass out pulls $x_{cm}$ toward it. · 质量加权平均值跟随质量——将质量移出会将$x_{cm}$拉向它。
Masses $2\ \text{kg}$ at $x=0$, $2\ \text{kg}$ at $x=2$, and $4\ \text{kg}$ at $x=5$. Find $x_{cm}$ (in m). · 质量$2\ \text{kg}$位于$x=0$，$2\ \text{kg}$位于$x=2$，且$4\ \text{kg}$位于$x=5$。求$x_{cm}$（单位：m）。
$x_{cm} = \dfrac{2(0)+2(2)+4(5)}{2+2+4} = \dfrac{24}{8} = 3\ \text{m}$.
Center of mass by integration
- For a continuous object, replace the sum with an integral 积分 over its mass:
- Slice the object into tiny bits $dm$, weight each by its position, and add them up.
用积分求质心
- 对于连续物体,把求和换成对质量的积分:
- 把物体切成微小的小块 $dm$,每块按它的位置加权,再全部相加。
Centre of mass of a system · 系统的质心
Tap the parts to see where the centre of mass of a two-body system sits. · 点击各部分查看双体系统质心的位置。
For a continuous body the sum in the center-of-mass formula becomes an . · 对于连续介质，质心公式中的求和变为一个。
$x_{cm} = \tfrac{1}{M}\int x\,dm$ -- an integral over the mass distribution. · $x_{cm} = \tfrac{1}{M}\int x\,dm$——对质量分布的积分。
Only external forces move it
- The center of mass accelerates only in response to the net external force 净外力.
- Internal forces -- however violent -- can never shift the center of mass.
- That is why the tumbling wrench's center of mass follows the same smooth path a simple thrown ball would.
只有外力才能移动它
- 质心只在净外力的作用下才加速。
- 内力——无论多么剧烈——永远无法移动质心。
- 这就是为什么翻滚扳手的质心,会沿着和一个简单抛球一样平滑的路径运动。
A hidden explosion inside a floating system (no outside forces) can shift its center of mass. · 漂浮系统内部的隐藏爆炸（无外力）可以移动其质心。
Explosion forces are internal, so they cannot move the center of mass. Only a net external force can. · 爆炸力是内部的，因此无法移动质心。只有净外力才能做到。
A $2\ \text{kg}$ mass sits at $x = 0$ and a $4\ \text{kg}$ mass at $x = 3\ \text{m}$.
- $x_{cm} = \dfrac{2(0) + 4(3)}{2 + 4} = \dfrac{12}{6} = 2\ \text{m}$.
- The center of mass sits closer to the heavier $4\ \text{kg}$ mass, as expected.
一个 $2\ \text{kg}$ 的质量位于 $x = 0$,一个 $4\ \text{kg}$ 的质量位于 $x = 3\ \text{m}$。
- $x_{cm} = \dfrac{2(0) + 4(3)}{2 + 4} = \dfrac{12}{6} = 2\ \text{m}$。
- 质心离更重的 $4\ \text{kg}$ 质量更近,正如所料。
The center of mass is not always the geometric center. For a uniform rod it is the middle, but load one end and the center of mass shifts toward the heavy end -- it is the mass-weighted average, not the shape's midpoint.
质心不一定是几何中心。对于均匀的杆,它就是中点,但如果一端加重,质心就会偏向重的那端——它是按质量加权的平均,而不是形状的中点。
The center of mass is the mass-weighted average position: $x_{cm} = \dfrac{\sum m_i x_i}{\sum m_i}$ (or $\tfrac{1}{M}\int x\,dm$ for a continuous body). A whole system moves as if all its mass were at that point, and only the net external force can accelerate it.
质心是按质量加权的平均位置:$x_{cm} = \dfrac{\sum m_i x_i}{\sum m_i}$(连续物体则为 $\tfrac{1}{M}\int x\,dm$)。整个系统运动起来就好像全部质量都在该点,而且只有净外力才能使它加速。