Center of Mass

62 Center of Mass

In this chapter we introduce an incredibly useful concept, namely that of the center of mass. The center of mass refers to the average position of all particles in a system, weighted by the mass,

$\vec r_{\rm com} = \Large \frac{m_1 \vec r_1 + m_2\vec r_2 + ... + m_N \vec r_N}{m_{\rm tot}},$

where $m_{\rm tot}$ is the total mass, $m_{\rm tot} = m_1 + m_2 + ... + m_N$ .

But why on Earth would one define such a thing? For one thing, if we are given a system that is comprised of many particles, and we wanted to assign that system a unique position, it seems like the center of mass would be a good choice. The more interesting reason, though, has to do with how the center of mass moves, namely

The center of mass of a system moves as though it is a single particle of mass $m_{\rm tot}$ that is subject to the total external force $\vec F_{\rm ext}$ .

This sentence above is amazing. For instance, if you had a rocket full of aliens all dancing the cha cha, the motion of the system as a whole is incredibly complicated to describe. But- if nothing is pushing on that rocket, the center of mass of the rocket + aliens will move with a constant speed. In fact, you’ll notice that in this book we have been treating real world objects as a single particle. This sentence justifies why: what we were really doing was describing how their center of mass moves.

It’s hard to appreciate just how awesome this without a good demo, so check this one out! (Original video here).

You can see that the motion of the system as a whole moves in a very complicated way. However, the center of mass just moves along at constant speed. Beautiful!

Now that we know why the center of mass is a useful concept, let’s go ahead and proving the sentence we wrote above, i.e. the center of mass moves as though it were a single particle subject to the total external force. We start out by computing the center of mass velocity.

Exercise 42.1: Center of Mass Velocity

A. Compute the velocity of the center of mass, $\vec v_{\rm com} \equiv d\vec r_{\rm com}/dt$ , and relate it to the system’s total momentum.

B. Summarize your final result as one or two sentences in English.

This suggests that for the purposes of momentum, we can imagine replacing a complicated system with many interacting particles by a single particle of mass $m_{\rm tot}$ . But is this really true when we also have external forces? Let’s find out.

Exercise 42.2: Center of Mass Acceleration

A. Find the acceleration of the center of mass.

B. Summarize your final answer as one or two sentences in English.

There it is:

The center of mass of a system moves as though it is a single particle of mass $m_{\rm tot}$ that is subject to the total external force $\vec F_{\rm ext}$ .

In the next chapter, we will go about using the concept of center of mass to solve some problems.

Key Takeaways

62 Center of Mass

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