# Centre of mass of particles

My question regarding centre of mass is less of a mathematical type and is more a conceptual one. As I have been taught that COM is a point (maybe within or outside the body) where crudely speaking all the mass of the body can be said to be concentrated and hence Newton's laws of motion can be applied as they are applicable only for point sized objects.

Then they taught me the derivations of COM of various objects and other related terminologies. This is where the trouble began. Till date I haven't been able to figure how am I to apply this concept (besides using it to apply Newton's laws of motion). Besides what do it means when we say velocity, acceleration or displacement of COM?

The confusion and bafflement intensified even more as they began using this concept in spring block problems to find maximum extension of the spring, maximum velocity of blocks, etc. The prof told us that the spring and the block system will perform both oscillatory and translational motion, the latter of which can be attributed to the velocity of COM (external force on the system of block and spring was 0).

So that means the velocity of centre of mass gives information only about the translational motion of the body or put it in another words does the body move with the velocity of COM? What about the oscillatory motion? Is that not manifested in the velocity of COM? Maybe, my whole notion of COM is screwed up.

I would love to have an explanation from beginning if anyone pleases to do so. Any help would greatly be appreciated.

• I think you are on the right track. The key statement is "So that means the velocity of centre of mass gives information only about the translational motion of the body." I suggest view this video from MIT first, and then edit and clear up you question by making a little bit more specific. Maybe provide a diagram of the problem that confused you. – ja72 Sep 5 '18 at 17:53

Okay, I'll try a different approach. I'll explain it as I wish it had been explained to me.

The key idea is much less elegant than it seems, but it is powerful anyways:

The position of centre of Mass (CM) is the (weighted) average radius of all particles.

Let's take the simplest case: two particles (A and B) of equal mass. The CM is located at its midpoint, just like our intuition tells us.

$\vec{r}_{CM}=\frac{\vec{OA}+\vec{OB}}{2}$

Now, what if particle "B" has twice the mass of A? Then B "pulls" the CM towards itself. The more mass, the more "attraction". Intuitively, we expect the CM to be closer to B.

What we do is, again, using the weighted average. $A$ contributes with 1/3. $B$ contributes with 2/3.

$\vec{r}_{CM}=\frac{1}{3}\vec{OA}+\frac{2}{3}\vec{OB}$

It's "like if B were two particles in one" $\vec{r}_{CM}=\frac{\vec{OA}+\vec{OB}+\vec{OB}}{3}$. Exactly the same result.

So, generalising this. A particle with mass $m_i$ contributes iwht $m_i/M$, where $M$ is the total mass of the system. That's why the weighted average position is

$$\vec{r}_{CM}=\sum_i \frac{m_i}{M} \vec{r}_i =\sum_i \frac{m_i \vec{r}_i}{M}$$

All this is equivalent to the usual formula:

$$\vec{r}_{CM}= \frac{\sum_i m_i \vec{r}_i}{\sum_j m_j}$$

## Its's just the weighted average of all positions. Weighted by the masses.

So, after having this, then you come up with some proeprties. For example, you find out that all individual weights, added up together, can be expressed as a single resultant force acting on this very point, and not other, but this CM.

This, among other properties, makes this point special. You can also check that

• the CM as a reference framse sees the sum of distances = 0.
• The total momentum of the obejct is the same as $M\cdot \frac{d\vec{r}_{CM}}{dt}$, so it behaves as a single massive particle placed there.

And so on.

So the idea is that it's the "average position of the system", and it turns out to have interesting properties.

For every collection of particles, we can define a quantity called the "center of mass". Definitions also exist in the case of continuous distributions, such as a solid object. I will call generally the matter containing the particles/continuous distribution "the object". The different parts of the object can interact with each other through forces. We call these forces "internal forces". The different parts of the object can also interact with "stuff" that is not part of the object through other forces, and we call these "external forces".

All these definitions become useful because it can be proven using Newton's laws that the movement of the point in space we call the center of mass only depends on the external forces. All the internal forces somehow cancel each other. Therefore, this point is space, the center of mass, obeys Newton's second law with the mass equal to the mass of the object. This is very useful as this allows extension of Newton's laws from point object to "real" objects. In a sense, if you define the "movement of the body" as being the movement of the center of mass, then yes, you can say that the body moves with the velocity of the center of mass. In other words, if you look at the body from a large distance, its movement will be identical to the movement of an object with all its mass concentrated in the center of mass.

Returning to a closer view, in the case of objects that don't have a rigid shape, such as blocks linked by a spring, the blocks will move relative to the center of mass. To see how the blocks move relative to the center of mass, you only have to take into consideration the internal forces, as the external forces are taken into account with the motion of the center of mass. This can simplify the solution to some problems. Simply look at the object and presume that there are no external forces, and solve Newton's equations. Then, to have the motion of the various parts of the object relative to "the world", add the movement of the center of mass.

As an exercise, you can consider two identical masses vertically aligned, linked by a spring of negligible mass falling in a uniform gravitational field. First, you can try to solve the problem using the center of mass concept. You see that the masses, by symmetry, will oscillate about the center of mass with an harmonic movement. The center of mass will also accelerate with a constant acceleration. To have the motion of each mass, simply add the position of the center of mass to the position of the mass relative to the center of mass. Now, solve the problem using the equations of motion for each mass. Therefore, there will be two forces on each mass: the gravitational force, and the spring force. The latter will couple your equations of motion, as it depends on the difference in position between the two masses. If you solve these equations, you will find the same solution as in the first case, but this will require more work.