In these problems, you will compute and analyze forces on many Argon particles under different initial conditions. Argon particles $($atoms$)$ are modeled as spheres of diameter $=0\mathrm{.}3405{10}^{-9}$ meters and mass $m=6\mathrm{.}634{10}^{-26}$ kg$.$ The interaction between a pair of Argon particles that generates the desired effect of particles coming together, sticking, and bouncing off is modeled using the Lennard$-$Jones $($LJ$)$ potential energy and force functions with the $LJ$ bond energy strength parameter $lon=1\mathrm{.}656{10}^{-21}$ Joules.

The code for simulating dynamics of many Argon particles adopts the following system of reduced units: lengths $($distances$)$ are measured in units of Argon atom diameter $,$ mass is measured in units of Argon atomic mass $m,$ and energy is measured in units of the LJ bond energy $lon.$ To convert distance $r^{**}$ in reduced units to distance $r$ in SI units $($meters$),$ the equation is $r=r^{**}[r]=r^{**},$ because the reduced unit of distance is $[r]=.$ A similar process is followed to convert other physical quantities in reduced units to SI units.

Use dimensional analysis and the reduced units of energy and length to get the reduced unit of force $f,$ that is$,$ find how many Newtons of force is $1$ reduced unit of force?

Using the reduced unit of force, $f,$ found in part $1,$ write an equation that relates the force $f^{**}$ in reduced units to the force $f$ in SI units $($Newtons$)?$ Its an equation similar to the equation noted above which connects distance in reduced units and SI units. This equation will come in handy in the following problem.

In the code, make $2$ Argon particles: one particle is at the origin $(0,0,0)$ and another one at $(1,0,0).$ So the distance between the particles $($in reduced units$)$ is $r^{**}=1.$ Run the code. The code prints out the initial vector force on each particle, showing explicitly the $x,y,z$ components of the force.

$($a$)$ What is the direction of the force on the particle at the origin: toward the other particle or away from it$?$

$($b$)$ Compute the magnitude of the force experienced by the Argon particle at the origin in reduced units $($note that we are asking for the magnitude, not individual components$).$

$($c$)$ What is this force in SI units $($Newtons$)?$