# B How to apply Newton's laws to non-rigid bodies?

1. Nov 6, 2016

### Prem1998

Suppose we have a body of finite size which can be translated, rotated, compressed and stretched. We have every information about the body. We have its dimensions and everything. We have its mass and density. But, since I'm here talking about a completely generalized version, so we have variable density. So, we have the density function. And, at last, we have the both the magnitude and direction of all the forces acting on the body along with the co-ordinates of the points on which each force is acting.
Then, how to predict the motion of the body? We can't draw free body diagrams for each point mass in the body, right?
EDIT: I think we can't have a force which acts for an instant and ceases to exist after that. Since, forces cause motion, so there must be an impulse transfer for a finite time. So, we also have the time for which the forces act.
One more thing, to deal with compression and stretching, we also have the strength of bonds between atoms of the body in terms of binding energy and also the force between any two atoms as a function of distance between them along with the range of that force. How to predict the behavior of this body in response to forces?

Last edited: Nov 6, 2016
2. Nov 6, 2016

### Nidum

Search on ' Continuum mechanics '

3. Nov 6, 2016

### Prem1998

Thanks, that will do.

4. Nov 6, 2016

### Staff: Mentor

You could, in principle, do that. This is essentially the approach taken by finite element analysis. For a relatively simple structure like you describe, it would be enough to just about it up into two pieces of variable length and write the forces as a function of length. This is how a beam is usually analyzed.

5. Nov 7, 2016

### Staff: Mentor

This is pretty much exactly what is done in deriving the general differential equations of motion for a deformable solid or a fluid. You identify a small differential volume in space (in which the fluid or solid is present) and do a mass balance and a momentum balance on the material entering, exiting, and accumulating within the small volume. You end up with the so-called continuity equation (i.e., mass balance) and the so-called equation of motion (i.e., momentum balance). These are combined with the deformational mechanics equation of the material (e.g. Hooke's law in 3D for a linearly elastic solid or Newton's law of viscosity in 3D for a Newtonian fluid). The latter leads to the so-called Navier-Stokes equation.

6. Nov 8, 2016

### mridul

Consider all mass to be situated on a single point is center of mass then apply law for that point

7. Nov 8, 2016

### Staff: Mentor

Do you really think this would tell you the kinematics of the deformation distribution experienced by a fluid or a solid?