4. In the rotational system shown below J = 1 kg. m, B=8 N.5m, and K = 15 N.m. The state variable of interest is the angular velocity of the drum, oriented in the same direction as the angular displacement. The initial position is, 8(0) = 6, and the initial angular velocity is w(0) =w. The applied torque is in the same direction as the displacement. J T() B XXXXXX a. Fiud an integro-differential equation for the velocity. Hint, in the spring term you can express the position as 0 =0(0)+ (dx. b. Find (s) in the frequency domain in terms of T.(s) and the initial conditions. e. Find the zero input response, 2.(8) in the frequency domain d. Find the zero input response, wi() in the time domain e. Find the zero state response, 12.(s) in the frequency domain when Te(t) = 1885 377 cos 21 fort > 0 f. Find the zero state response, West) in the time domain. g. Find the total response in the time domain. 4. In the rotational system shown below J = 1 kg. m, B=8 N.5m, and K = 15 N.m. The state variable of interest is the angular velocity of the drum, oriented in the same direction as the angular displacement. The initial position is, 8(0) = 6, and the initial angular velocity is w(0) =w. The applied torque is in the same direction as the displacement. J T() B XXXXXX a. Fiud an integro-differential equation for the velocity. Hint, in the spring term you can express the position as 0 =0(0)+ (dx. b. Find (s) in the frequency domain in terms of T.(s) and the initial conditions. e. Find the zero input response, 2.(8) in the frequency domain d. Find the zero input response, wi() in the time domain e. Find the zero state response, 12.(s) in the frequency domain when Te(t) = 1885 377 cos 21 fort > 0 f. Find the zero state response, West) in the time domain. g. Find the total response in the time domain