Problem $1(12$ points$)$

Start from a free$-$body diagram of all forces applied to the mass $($external force $F$ and

reaction force of all elements connected to the mass$),$ to derive the equations of motion

$($a differential equation$)$ that relate the force F to the velocity $v=x^{}$ for the systems below,

as a function of parameters of the spring $($stiffness k$),$ mass $($inertia m$),$ and dashpot

$($viscous friction c $).$

For both systems, determine an equivalent electrical circuit that yields the same equations

of motion when converting variables from one domain to another using the through$/$across

variable analogy. The equivalent electrical circuit will model the response of the system

to a current input $($like the mechanical system is subject to a force input$).$ Show all steps,

assumptions, and determine the relationships between electrical and mechanical

elements.

$($a$)$