$1$ Extra Credit Challenge

Learning Objective: $1)$ Apply Lagrange's equations to determine equations of motion.

Given: The structure shown below

Initial values are: $x[0]=x_0=0\mathrm{.}25m$ and $x^{}[0]=x_0^{}=0\frac{(m)}{(s)},[0]={}_0=\frac{}{6}$rad and ${}^{}[0]={}_0^{}=0\frac{\text{r}\text{a}\text{d}}{s}$

Constants: $m_1=1kg$;$m_2=4kg$;$m_3=3kg$

Note: You may treat $m_2$ as a point mass $($not a sphere$).$

$k_1=50\frac{N}{m}$;$k_2=5\frac{N*m}{\text{r}\text{a}\text{d}}$

$c1=16\frac{N*s}{m}$;$c2=4\frac{N*m*s}{\text{r}\text{a}\text{d}}$

$L=2\mathrm{.}5m$

Find: $1)$ Draw a vector to each the fixed point $($wall$)$ to each Center of Mass to help solve for the absolute velocity of

each COM. $2)$ Write the vector in terms of the variables in question. $3)$ Take the derivative of each vector.

Determine the kinetic energy $($T$)$ for each component. $5)$ Determine the potential energy $($V$)$ of each

component. $6)$ Determine the energy dissipation term $($R$)$ for each component. $7)$ Equations of motion $($for

$x(t)$ and $(t)$ using Lagrange's Equations.

Sol'n: