# Deformation of a body and centre of mass acceleration

Let's take a flexible body like clay, and we apply force $$F$$, on it such that, some part of it deforms, under action of force.

When deformation is taking place, can we say that acceleration of centre of mass of system of clay body is at rest?

Technically, what I thought was that centre of mass should gain an acceleration, becaus, some particles of our body is moving, while some are at rest, so we can write:

$$a_{cm}=\frac{\sum_i m_ia_i}{\sum_i m_i}$$

where $$m_i$$ is mass of individual particles gaining acceleration while deformation takes place.

Writing and summing for other particles which don't participate in deformation is useless, since they $$a_i=0$$.

But answer in my book, is given as $$a_cm=0$$ since, body is at rest as whole, which confuses me even more. Help and tell any flaws in my understanding, if found any. Any help is massively appreciated.

The book seems to be wrong, unless it's talking about a symmetrical force that presses from both sides.

If a force is pushing from one side and deforms the clay, then object that is pushing (let's assume for simplicity that it doesn't deform) experiences the same force, in the opposite direction (from Newton's 3rd law).

This would cause a change in momentum of the pushing object equal to $$Ft$$, force x time. From conservation of momentum, the clay (on the whole) must gain the same momentum in the opposite direction. Since it's mass isn't changing, the COM must acquire a velocity and must have been accelerated.

As viewed from outside both the clay and the pushing object, the total momentum of the system is then conserved. The COM of the whole system doesn't accelerate, but the COM of the clay does.

• @Jhon Hunter that's the concept I gave in my comment. Oct 14 at 10:37

The acceleration of centre of mass only depends on the net external force acted upon the body. Internal forces cancel each other (newton's 3rd law) and do not contribute to centre of mass acceleration. So your applied force and the deformation forces produced by the force cancel with each other.

$$\mathbf{a}_{com} = \frac{\mathbf{F}_\text{ext net}}{\text{total mass}}$$

In your case, the clay is at rest, which implies net external force is 0, so acceleration of centre of mass is 0.

• Thanks for answering. But can you point out error in my equation/understanding? Oct 14 at 7:36
• There is no error in your equation. Here you are taking the individual particles of the clay as the particles of the system (as it looks like in your post) and that's the mistake. The particles of the body-clay system will be the centre of mass of clay and the centre of mass of the force applying body. Since the deformation of clay would cause the acceleration in its centre of mass but to compensate this the force applying body must also move so that that centre of mass of the clay body system remains at rest as said in the above answer. Oct 14 at 7:48