#$2(25$ points$)$

Five springs are arranged between two end points, as shown. $($Remember that the vertical "connection

bars" don't tip; this is a problem of one$-$dimensional motion.$)$ The springs are individually well described

by Hooke's constitutive laws of the form $f_1=k_1{x}_1,$ etc.

$2-1.(5$ pts $)$ Write down the force balance equation for the connection labeled $C.$

$2-2.($True$/$False; $2$ pts each$)$ Which of the following are correct geometric continuity equations if the

points A and D are fixed $)=v_D=(0?$ A statement is false if it is not necessarily true.

a$)v_2+v_3=v_4+v_5$

b$)v_2=v_3$

c$)v_1+v_3+v_4+v_6=0$

d$)v_1+v_2+v_3+v_6=0$

e$)v_1+v_5=v_2+v_6$

$2-3.(5$ pts $)$ Write a correct geometric continuity equation involving $v_5$ if the absolute velocities of points

A and $D$ are equal to $v_A$ and $v_D,$ respectively.

$2-4.(5$ pts$)$ Consider the ensemble all together as one device, with forces applied at the left and right

ends. Model the ensemble as a single Hooke's law spring, with spring constant $k_{\text{r}\text{e}\text{d}\text{b}\text{o}\text{x}}$ and find

$k_{\text{r}\text{e}\text{d}\text{b}\text{o}\text{x}}$ in terms of the five known spring constants. Hint: use the parallel and series results

developed in Problem $1.$ As always, show your derivation, don't just give an answer.