$7\mathrm{.}6.$ A velocity input $v_a(t)$ is applied to point $A$ in the mechanical system shown in Figure P$7\mathrm{.}6.$

a$.$ Write the system's differential equation in terms of the velocity $v_1.$

b$.$ What is the time constant $$ for the system?

c$.$ Sketch the response when $v_a(t)=0$ for $t0$ and $v_1(0)=10.$

d$.$ Repeat parts $($a$),($b$),$ and $($c$)$ when the velocity input is replaced by a force $f_a(t)$ applied at point $A,$ with the positive sense to the right. Explain why the expression for $$ differs from the answer to part $($b$).$