Problem $2$

For the mechanical system below, $M=2kg,K_2=5\frac{N}{m},B=3N-\frac{s}{m},$ and $J_1=10kg-m^2.$ The

external torque, $T_a(t),$ is a step input with a magnitude of $10N-m.$

$($a$)$ Using Newton's Laws, determine the equations of motion for the system.

$($b$)$ Using the results from $($a$),$ apply the Laplace Transform to each equation.

$($c$)$ Using the results from $($b$),$ solve for the mass displacement in terms of the $\text{'}$s$\text{'}$ variable, $x(s).$ How

many poles and zeroes does the expression of $x(s)$ have? Justify your answer.

Note:You do not need to apply the inverse of the Laplace Transform.