$F=m*a$ accounting for pressure wave propagation

Imagine a long deformable rod which has just been hammered on the top end (the bottom end is clamped to Earth). Consider a time interval $$dt$$ = $$t_{2}$$ - $$t_{1}$$ in which the pressure wave is travelling somewhere within the length of the rod (meaning some portion of the object has already "felt" the impact whereas the remaining portion is still at rest (assuming the whole rod was at rest in the start)).

My question concerns how to apply $$F=M*a$$.

Specifically, I am using the following equation: $$F_{net}|t_{2}*dt = (M*v)|dt$$ where $$F_{net} = F_{impact} + M*g - F_{K*du}$$ and $$F_{impact}$$ represents the hammer impact force, $$K$$ is the stiffness of the rod and $$d_{u}$$ is the relative displacement between the top end and the end of the affected part of the rod (== displacement of the top end since the displacement at the end of the affected part of the rod = 0).

For my example, the mass density of the rod decreases with height, say linearly. My question is what $$M$$ and $$v$$ should I use:

1. The affected mass, i.e. the mass of the part of the rod which the pressure wave already travelled, at time instants $$t_{2}$$ and $$t_{1}$$, and the corresponding velocities? Here masses will differ as well as the velocities. $$F_{net}|t_{2}*dt = (M*v)|t_{2} - (M*v)|t_{1}$$
2. The affected mass at time instant $$t_{2}$$ and the velocities at time instants $$t_{2}$$ and $$t_{1}$$? Here mass will be the same, multiplied by the velocities' values existing at time instants $$t_{2}$$ and $$t_{1}$$ for the affected length of the rod at $$t_{2}$$? $$F_{net}|t_{2}*dt = M|t_{2}*(v|t_{2}- v|t_{1})$$ It's not quite like this because a part of the $$M|t_{2}$$ has zero velocity at $$t_{1}$$ but it was easier to write it like this.
3. The affected mass at time instant $$t_{2}$$ and the velocities at time instants $$t_{2}$$ and $$t_{1}$$? Here mass will be the same, multiplied by the velocities' values existing at time instants $$t_{2}$$ and $$t_{1}$$ ? $$F_{net}|t_{2}*dt = M|t_{2}*(v|t_{2}- v|t_{1})$$ This is 2) but here we simply multiply the mass at $$t_{2}$$ with the velocity at $$t_{1}$$, i.e. we neglect the fact that at $$t_{1}$$ part of the mass at $$t_{2}$$ has zero velocity.
4. Other?

This is a simplification of a problem I've been working on trying to come up with an analytical model to study MDOF collisions between deformable bodies. I want to simulate what happens during impact, I want to go beyond the simple determination the average impact force.

$$m$$ is the mass of your “system” and $$v$$ is the velocity of the center of mass of the system. The system could be the whole rod or some subset of the rod. The easiest is the whole rod.

If it is a subset of the rod, like in your 2, then you need to consider the force between the system and the rest of the rod. If the rod were in free fall then that force could be 0.

Your 1 doesn’t work because in that case the system’s momentum has changed not only because of the force but also because of the changing definition of the system. This can be done, but it is an advanced concept.

• THanks for the reply. I added a bit more text to my post. Can you please recheck it? For the free fall example, say a long rod hitting the ground, at time t2 (> t1) the mass should be left and right of the F=M*a the Mass at this time instant and not the Mass for t1 or t2 right? Thanks Commented May 14, 2022 at 23:22
• @jpcgandre The easiest system to use is the entire rod. It is a fairly complicated topic to use different masses at different times. Definitely not just F=ma. I would strongly recommend against using different masses at different times
– Dale
Commented May 15, 2022 at 1:42
• I want to simulate what happens during impact, I want to go beyond the simple determination of the average impact force. So I need to use different masses at different times. That's well understood for my part. The reason for this post was to see for a specific time instant and time interval which values of mass and velocities to use... Commented May 15, 2022 at 11:57
• Also my option 2) applied to this example will lead to (M*v)|t2 being possibly quite different than for |t1 since part of the rod's mass affected at t2 is not affected yet at t1, so even if I plug in the same mass M at t2 and t1 the result will end up equal to use different masses since part of M|t2 will be multiplied by 0. Commented May 15, 2022 at 12:00
• Can you give more information on your assessment of my case or provide any relevant references, similar examples? Thanks! Commented May 15, 2022 at 12:00