Post by Richard Livingston Post by Luigi Fortunati
In my animation
Regarding my animation, I based myself on the following considerations=
which seem correct to me.
The stone is a set of billions of billions of particles which, in my
animation, for simplicity, I have reduced to 5 (from A to E).
Between C and D the forces are of magnitude 3, between B and C of
magnitude 2 and between A and B of magnitude 1.
Always for the third principle.
A more precise way to model this, the way it is done in computer codes
of elastic collisions, is to model the material as a fine array of mass=
Post by Richard Livingston
connected to adjacent masses by short stiff springs. When the point
E contacts F there is actually a short spring that reduces the theoreti=
Post by Richard Livingston
instantaneous stop of E to a rapid deceleration.
As E comes to a stop the spring to D compresses and causes D to slow
down, then C then B, etc.
The impulse from E colliding with F propagates back through the mass
(D to C to B to A etc) at the speed of sound in that material.
I agree with you: the impulse of the collision goes leftward from
particle E to A but not with the same intensity.
Is it correct to say that the exchange of forces decreases in the
proportion of 5 between F and E, 4 between E and D, 3 between D and C,
2 between C and B and 1 between B and A?
[[Mod. note --
1. You seem to be using the word "impulse" in a way different from how
it's used in physics ( https://en.wikipedia.org/wiki/Impulse_(physics) ).
2. I don't quite know what you mean by the term "exchange of forces".
But if we interpret your question as asking if the forces
F_FE = the force F exerts on E
F_ED = the force E exerts on D
F_DC = the force D exerts on C
F_CB = the force C exerts on B
F_BA = the force B exerts on A
are in the ratio 5:4:3:2:1, then we can apply Newton's 2nd law to try
to find out. To do this, let's introduce a bit of terminology: Let's
* we'll call the entire ball (the combined system A+B+C+D+E) "ABCDE"
* we'll call the combined system A+B+C+D "ABCD"
* we'll call the combined system A+B+C "ABC"
* we'll call the combined system A+B "AB"
Now we observe that
* the net (horizontal) force acting on ABCDE is F_FE, i.e.,
F_FE is the only (horizontal) external force acting on ABCDE
* the net (horizontal) force acting on ABCD is F_ED, i.e.,
F_ED is the only (horizontal) external force acting on ABCD
* the net (horizontal) force acting on ABC is F_DC, i.e.,
F_DC is the only (horizontal) external force acting on ABC
* the net (horizontal) force acting on AB is F_CB, i.e.,
F_CB is the only (horizontal) external force acting on AB
* the net (horizontal) force acting on A is F_BA, i.e.,
F_BA is the only (horizontal) external force acting on A
* the net (horizontal) force on A is F_BA
If we assume A, B, C, D, and E to all have the same mass m, then
* the mass of ABCDE is m_ABCDE = 5m
* the mass of ABCD is m_ABCD = 4m
* the mass of ABC is m_ABC = 3m
* the mass of AB is m_AB = 2m
* the mass of A is m_A = m
Since the masses m_ABCDE:m_ABCD:m_ABC:m_AB:m_A are in the ratio
5:4:3:2:1, by F_net=ma the net forces will be in the same ratio
if and only if the accelerations of the center-of-masses of
ABCDE, ABCD, ABC, AB, and A are all the same. But in reality these
accelerations *won't* all be the same -- the springs will compress
by different amounts under the different forces, so the balls won't
all accelerate at the same rate, and the center-of-mass accelerations
of ABCDE, ABCD, ABC, AB, and A *won't* all be the same. So, we
conclude that the forces F_FE:F_ED:F_DC:F_CB:F_AB *aren't* in the
ratio 5:4:3:2:1. Figuring out the actual (time-dependent) forces
requires a more careful analysis of the dynamics of the collision.