AP Physics 1 · Unit 4 · Topic 5 of 6 · 5 min lesson

Center of Mass

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.

Quiz this topic · 5 questions Watch the review video ↗

The big idea

The center of mass is the mass-weighted average position of a system: x_cm = (m₁x₁ + m₂x₂) ÷ (m₁ + m₂). It sits closer to the heavier object, and it moves as if all the mass were there: only an external force can change its velocity.

1. Finding it

x_cm = [(2)(0) + (3)(5)] ÷ 5 = 3 m: closer to the 3 kg ball.

x_cm = Σmx ÷ Σmweighted average

Closer to the heavier massmidpoint only if masses are equal

a_cm = F_{net, ext} ÷ m_totalNewton’s second law for the system

2. Internal forces can’t move it

Forces between parts of a system come in third-law pairs, so they can’t change the system’s total momentum, or the motion of its center of mass.

With no net external force, the center of mass keeps moving at a constant velocity (or stays still).

A firework that explodes at the top of its arc sends pieces everywhere, but until the first piece lands, the center of mass of all the pieces keeps following the original parabola: gravity is still the only external force.

3. Example, then try it

Example

Three masses lie on a line: 1 kg at 0 m, 2 kg at 2 m, and 1 kg at 6 m. Where is the center of mass?

x_cm = [(1)(0) + (2)(2) + (1)(6)] ÷ (1 + 2 + 1) = 10 ÷ 4 = 2.5 m.

Try it · work it on paper first

Two skaters on frictionless ice hold the ends of a 5.0 m rope. Skater A (60 kg) is at x = 0; skater B (40 kg) is at x = 5.0 m. They pull themselves together. Where do they meet?

Center of mass: x_cm = [(60)(0) + (40)(5.0)] ÷ 100 = 2.0 m.

Forces: the rope’s pulls are internal, and the ice has no friction, so there’s no net external force.

So the center of mass stays put at 2.0 m, since the skaters started at rest.

They meet where the center of mass is: x = 2.0 m, closer to the heavier skater.

Answer: at x = 2.0 m: the lighter skater moves 3.0 m and the heavier one only 2.0 m.

4. Common mistakes

Using the midpoint.The center of mass is a weighted average. It’s only at the midpoint when the masses are equal.

“Internal forces move the center of mass.”Only external forces can change its velocity.

“It must be inside the object.”For a ring or a boomerang, the center of mass is in the empty space.

Check yourself

A person walks from one end of a canoe to the other on still water (ignore water drag). What happens to the center of mass of the person + canoe?

The forces between the person and the canoe are internal, so the center of mass can’t start moving. The canoe slides backward as the person walks forward.

Ready to practice?

5 questions on Center of Mass, each with a hint and an explanation for every wrong answer.

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