Systems and Center of Mass
Introduced| English | Meaning |
|---|---|
| center of mass/ˈsentə ɒv mæs/ | The mass-weighted average position of a system's constituent parts. |
| system/ˈsɪstəm/ | The object or collection of objects chosen for analysis, separated conceptually from its surroundings. |
| internal forces/ɪnˈtɜːnl ˈfɔːsɪz/ | internal forces |
| external forces/ekˈstɜːnl ˈfɔːsɪz/ | A force exerted on a chosen system by something outside that system. |
| point particle/pɔɪnt ˈpɑːtɪkl/ | point particle |
| integral/ˈɪntɪɡrəl/ | A mathematical quantity obtained by accumulation, represented by a definite integral, or a family of antiderivatives represented by an indefinite integral. |
| net external force/net ekˈstɜːnl fɔːs/ | net external force |
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.
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.
The centre of mass is the mass-weighted average position
Select all true statements about the center of mass.
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The first two are the core rules. The third is false -- it is the mass-weighted center, not the shape's midpoint.
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.
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)?
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$x_{cm} = \dfrac{1(0) + 3(4)}{1 + 3} = \dfrac{12}{4} = 3\ \text{m}$ -- close to the heavier mass.
Move one mass farther from the others. The center of mass...
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The mass-weighted average follows the mass -- moving a mass out pulls $x_{cm}$ toward it.
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).
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$x_{cm} = \dfrac{2(0)+2(2)+4(5)}{2+2+4} = \dfrac{24}{8} = 3\ \text{m}$.
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.
Centre of mass of a system
Tap the parts to see where the centre of mass of a two-body system sits.
For a continuous body the sum in the center-of-mass formula becomes an ____.
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$x_{cm} = \tfrac{1}{M}\int x\,dm$ -- an integral over the mass distribution.
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.
A hidden explosion inside a floating system (no outside forces) can shift its center of mass.
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Explosion forces are internal, so they cannot move the center of mass. Only a net external force can.
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.
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.
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.
Write from a centre of mass at 4 m
- A 1 kg mass is at x = 0. A 4 kg mass is at x = 5 m.
- Find the centre of mass.
A 1 kg mass sits at x = 0 and a 4 kg mass sits at x = 5 m. Where is the centre of mass, in metres?
20 divided by 5 is 4.
Where is the centre of mass?
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The mass-weighted average of 0 and 5 m is 4 m. Other options — 3 m (the lesson centre); 5 m (the position of the heavy mass).