# When and why are we allowed to treat a rigid body as a point mass?

When the subject Mechanics first taught, it is common that we explicitly state that the Newton's laws are valid only for point masses, and then we give examples of rigid bodies colliding with each other.

However, is there any argument that why we are allowed to treat a rigid bodies as point mass when the interaction between them "seems simple", as in the case of linear collision?

Edit:

I'm looking for an argument that is plausible to from the point of view of someone who is new to the subject.

• Yes. If no rotation is involved or if the moment of inertia of the rigid body is such that the rotational momentum is negligible compared to each of the applied moments. Commented Aug 4, 2018 at 4:34
• @ChesterMiller Ok, but that is an argument that you give after explaining all that staff, but this is done just in the beginning of the subject, so I'm looking for an argument that is plausible to from the point of view of someone who learns the subject at the moment.
– Our
Commented Aug 4, 2018 at 5:20

Well, one case is with regards to gravity. Since this force acts on all the parts of the object, it amounts to working on "all of it" so we can treat all of it as moving due to this force. This is especially so in the case of a uniform gravity (field), as is often taught initially (i.e. all free-falling bodies move with acceleration g). When it comes to gravity between spherical planets, one needs to invoke Newton's theorems that each such body generates and is affected by gravity like a point mass.

Another elementary case is collisions and motions in a straight line of balls (or, better, frictionless discs and so on). It is clear from the definition of momentum why the body will move in a straight line as a whole: each part moves on its own, so the whole body does. And collisions are momentary, so you can "ignore" the complex process of the collision itself and focus on the outcome. Whatever the total internal forces do during the collision, we're just interested in what happens after them to the motion of the whole ball and hence the motion of its center.

This really leaves only things like pulling, pushing, and friction as the more problematic still-elementary cases. A real explanation of these would require moments. However at the elementary level before reaching moments, one can say that we're ignoring the rotation of bodies at this stage. So consider for example a force pulling a block of wood horizontally. We can imagine the force pulling from the center of one end, and the block as being composed of a set of springs, so that the force stretches all springs throughout the body until the entire body is pulled. Which I think makes it intuitive that the force is pulling all the mass, and hence we can track the block by tracking it as a point mass.

I think the above should justify things enough for most cases encountered in a basic Newtonian mechanics course, before it reaches rotational energy, moments, and so on.