Systems and Center of Mass · 系统与质心
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| centre of mass/ˈsentə ɒv mæs/ | 质心 | zhì xīn |
| system/ˈsɪstəm/ | 系统 | xì tǒng |
| external force/ekˈstɜːnl fɔːs/ | 外力 | wài lì |
| internal force/ɪnˈtɜːnl fɔːs/ | 内力 | nèi lì |
| particle/ˈpɑːtɪkl/ | 质点 | zhì diǎn |
A wrench tumbles — yet one point glides straight
- Slide a spinning wrench across ice and every part traces a messy loop.
- But one special point drifts in a perfectly straight line — the centre of mass 质心.
- Treat a complicated object (or a group of objects) as if all its mass sat there.
- That single idea turns a whole system 系统 into one easy-to-track point.
扳手翻滚——却有一个点直线滑行
- 让旋转的扳手在冰面上滑行,它的每一部分都划出杂乱的环线。
- 但有一个特殊的点沿着完美的直线漂移——质心。
- 把一个复杂的物体(或一组物体)当作全部质量都集中在那一点。
- 这一个想法就把整个系统变成一个易于追踪的点。
What a system is
- A system is any group of objects you choose to study together.
- Everything else is the surroundings; forces from outside are external forces 外力.
- Forces between members of the system are internal forces 内力.
- By Newton's third law, internal forces come in canceling pairs — they never move the centre of mass.
什么是系统
- 系统是你选择一起研究的任意一组物体。
- 其余的一切是环境;来自外部的力是外力。
- 系统内部各成员之间的力是内力。
- 由牛顿第三定律,内力成对相消——它们绝不会移动质心。
By Newton's third law, the internal forces of a system always ____ in pairs. · 根据牛顿第三定律，系统的内力总是成对地____。
Internal forces are action–reaction pairs, so they sum to zero and cannot move the centre of mass. · 内力是作用力与反作用力对，因此它们的总和为零，不能移动质心。
Locating the centre of mass
- The centre of mass is the mass-weighted average of every part's position.
- For point masses on a line: $\displaystyle x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2 + \cdots}{m_1 + m_2 + \cdots}$.
- It sits closer to the heavier mass — the big mass "pulls" the average toward it.
- It need not be inside the object: a ring's centre of mass is in the empty hole.
确定质心的位置
- 质心是每一部分位置的质量加权平均。
- 对于一条线上的质点:$\displaystyle x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2 + \cdots}{m_1 + m_2 + \cdots}$。
- 它更靠近较重的质量——大质量把平均值"拉"向自己。
- 它不一定在物体内部:圆环的质心在空心处。
Find the balance point · 找到平衡点
Change the two masses and their distances, and see where the system balances — the centre of mass. · 改变两个质量及其距离，观察系统在哪里平衡——即质心位置。
A $2\ \text{kg}$ mass is at $x = 0$ and a $6\ \text{kg}$ mass at $x = 8\ \text{m}$. Where is the centre of mass, in metres? · 一个$2\ \text{kg}$的质量位于$x = 0$处，另一个$6\ \text{kg}$的质量位于$x = 8\ \text{m}$处。质心在何处，单位为米？
$x_{\text{cm}} = \dfrac{2(0) + 6(8)}{2 + 6} = \dfrac{48}{8} = 6\ \text{m}$ — closer to the heavier mass. · $x_{\text{cm}} = \dfrac{2(0) + 6(8)}{2 + 6} = \dfrac{48}{8} = 6\ \text{m}$——更靠近较重的质量。
The centre of mass of a system is best described as the: · 系统的质心最准确的描述是：
It is · 它是 $\frac{\sum m_i x_i}{\sum m_i}$ — each position weighted by its mass. · 它是$\frac{\sum m_i x_i}{\sum m_i}$——每个位置都乘以其质量权重。
Two equal · 相等 masses sit at $x = 2\ \text{m}$ and $x = 6\ \text{m}$. Where is their centre of mass, in metres? · 两个相等的质量分别位于$x = 2\ \text{m}$和$x = 6\ \text{m}$处。它们的质心在何处，单位为米？
Equal masses ⇒ the centre of mass is the midpoint: $(2 + 6)/2 = 4\ \text{m}$. · 质量相等 ⇒ 质心是中点：$(2 + 6)/2 = 4\ \text{m}$。
Why the centre of mass matters
- External forces alone move the centre of mass: $\vec F_{\text{net,ext}} = M\,\vec a_{\text{cm}}$.
- No matter how the parts spin or collide inside, the centre of mass obeys this simple law.
- A diver somersaults, but their centre of mass follows a smooth parabola.
- This is why we can model an extended object as a single particle 质点.
质心为何重要
- 只有外力才能移动质心:$\vec F_{\text{net,ext}} = M\,\vec a_{\text{cm}}$。
- 无论内部各部分怎样旋转或碰撞,质心都遵守这条简单的定律。
- 跳水运动员翻着筋斗,但他的质心沿着一条平滑的抛物线运动。
- 这就是为什么我们能把一个有大小的物体看成一个质点。
Select all · 所有 true statements about the centre of mass. · 选择所有关于质心的正确陈述。
It is mass-weighted (closer to heavy parts), moved only by external forces, and lets us treat the body as a point — but it is not always the geometric centre. · 它是质量加权的（靠近重部），仅由外力移动，使我们能将其视为点——但它并不总是几何中心。
The centre of mass is not always the geometric centre, and it need not lie within the material. For a doughnut or a boomerang it sits in empty space. Only for a uniform, symmetric object do the two coincide.
质心不一定是几何中心,也不一定落在材料内部。对于甜甜圈或回旋镖,它落在空处。只有对均匀、对称的物体,两者才重合。
The centre of mass must always lie inside the material of the object. · 质心必须始终位于物体的材料内部。
A ring or boomerang has its centre of mass in empty space. Only symmetric solid bodies keep it inside. · 圆环或回旋镖的质心位于真空中。只有对称的实心物体才使其保持在内部。
A $2\ \text{kg}$ ball sits at $x = 0$ and a $6\ \text{kg}$ ball at $x = 8\ \text{m}$.
- $x_{\text{cm}} = \dfrac{(2)(0) + (6)(8)}{2 + 6} = \dfrac{48}{8} = 6\ \text{m}$.
The centre of mass is at $6\ \text{m}$ — much closer to the heavier $6\ \text{kg}$ ball.
一个 $2\ \text{kg}$ 的球在 $x = 0$,一个 $6\ \text{kg}$ 的球在 $x = 8\ \text{m}$。
- $x_{\text{cm}} = \dfrac{(2)(0) + (6)(8)}{2 + 6} = \dfrac{48}{8} = 6\ \text{m}$。
质心在 $6\ \text{m}$——明显更靠近较重的 $6\ \text{kg}$ 球。
A system is a chosen set of objects; its centre of mass is the mass-weighted average position, $x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$, lying closer to heavier parts. Internal forces cancel in pairs; only external forces move the centre of mass ($\vec F_{\text{net,ext}} = M\vec a_{\text{cm}}$).
系统是一组选定的物体;它的质心是质量加权平均位置 $x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}$,更靠近较重的部分。内力成对相消;只有外力才移动质心($\vec F_{\text{net,ext}} = M\vec a_{\text{cm}}$)。