- #1

Jpcgandre

- 20

- 1

- TL;DR Summary
- Newton's second law and 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 traveling 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) Other?

A similar setting is the long rod hitting the ground after being dropped from a height $h$ above ground.

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) Other?

A similar setting is the long rod hitting the ground after being dropped from a height $h$ above ground.