Consider the system. v(t) B M XXXXXX T f(t) M = 10 kg, B=5 N-s/m, v(t) in m/s, f(t) in N Mdv(t) dt + Bv(t) = f(t) Consider the system described by the above differential equation where the velocity v(t) and the applied force f(t) are the output and the input, respectively, of the system. (1a) Convert the above differential equation into the form of the typical first-order dynamic system: dv(t) +v(t)= v(t), and explain the physical meaning of the two parameters 7 and v. (5%) dt (1b) According to the information of (1a) and the assumption of zero initial condition v(0) = 0 and unit step input f(t) = u(t), use the Laplace transform approach to find the analytic function v(t), which is called the step response of the system. (3%) (1c) Use MATLAB "plot" command to plot the analytic function v(t) obtained in (1b) on a graph versus time t. On the graph, specify the value of time constant 7 and the values of v(t) at t = 7, and at the steady state as t. (3%) (1d) According to the information of (1a) and the assumption of zero input f(t) = 0, and the initial state v(0) = v= 1 m/s, use the Laplace transform approach to find the analytic function v(t), which is called the initial state response of the system. (3%) (1e) Use MATLAB "plot" command to plot the analytic function v(t) obtained in (1d) on a graph versus time t. On the graph, specify the value of time constant 7 and the values of v(t) at t = 7, and at the steady state as t. (3%) dv(t) dt (1f) Consider the system with differential equation: 2 -+v(t)= 0.2 f(t). Find the transfer function of the system, G(s) = V(s)/F(s), where V(s) and F(s) are the Laplace transforms of v(t) and f(t), respectively. (3%) (1g) What is the characteristic equation of the system? (3%) (1h) Find the pole (or poles) of the system? (3%) (li) Describe the relationship between the system pole and the time constant 7? How do they affect the time response? (4%)