Newton's Second Law: F = ma relates the net force on an object to its acceleration. But what counts as "the object"? A car with passengers, two carts tied together, a whole train? In physics you get to choose: the things you include form the system, and everything else is the surroundings.
Key idea
Forces between parts of the system are internal; forces from outside are external. Only external forces change the motion of the system's center of mass, the mass-weighted average position: $x_{cm} = \dfrac{m_1x_1 + m_2x_2 + \cdots}{m_1 + m_2 + \cdots}$.
Why it works
Internal forces come in third-law pairs, equal and opposite, both inside the system, so they cancel when you add up all the forces on the system. What's left is the external force, and it moves the system as if all the mass sat at one point. That point is a weighted mean: like a fair share, but heavier parts get more say.
Worked examples
Worked example: Two masses
A $6\ \text{kg}$ block sits at $x = 0$ and a $2\ \text{kg}$ block at $x = 4\ \text{m}$. Where is their center of mass?
- $x_{cm} = \dfrac{(6)(0) + (2)(4)}{6 + 2} = \dfrac{8}{8}$.
- $x_{cm} = 1\ \text{m}$: a quarter of the way, close to the heavier block, not halfway.
Answer: $1\ \text{m}$
Worked example: Three masses
Masses of $2$, $3$ and $5\ \text{kg}$ sit at $x = 0$, $1$ and $3\ \text{m}$. Find $x_{cm}$.
- Top: $$(2)(0) + (3)(1) + (5)(3) = 18\ \text{kg}\cdot\text{m}.$$ Bottom: $10\ \text{kg}$.
- $x_{cm} = 1.8\ \text{m}$.
Answer: $1.8\ \text{m}$
Worked example: Choosing the system
A tow truck pulls a car with a chain. Which forces are internal if the system is the truck, the chain and the car?
- Internal: the chain's pull on the car and the car's pull back on the chain; the truck's pull on the chain and the chain's pull back. Each pair cancels within the system.
- External: gravity, the road's normal and friction forces on the tires, air drag. Only these change the motion of the whole rig.
Answer: The chain forces are internal; road, gravity and air forces are external.
Common mistake
Putting the center of mass halfway between two objects whatever their masses. Halfway is right only for equal masses. Check: the center of mass is always between the objects and closer to the heavier one; if one mass is much bigger, it sits almost on top of that mass.