HELP!!! ASAP!!! DIFERENTIAL EQUATION Case 2: We assume that the passenger is passive like in Case 1. However, this time, the damping mechanismis broken in such a way that the suspension only relies on the spring.g) Modify your answer in point a to account for the broken damper (c=0).h) Write down the characteristic equation of the differential equation in point g!i) Find the roots of the characteristic equation in point h! Use m=100 kg!j) Write down the position of the passengers x(t) if the passenger was initially at x(0) = 0 moving to the leftwith speed 10 m/s!k) Will the movement of the passenger stabilize over time (x(t) goes to zero)? If not, is it possible to make itstable by modifying the spring constant k?1) How long would it take for the passenger to pass through his/her initial position 10 times? (Do not countthe initial moment when x(0)=0)!Case 3: Now, with the damper still broken, the vehicle is experiencing a vibration in such a way that anexternal sinusoidal force F is experienced by the passenger where F = Acos(wt). Here, the amplitude Aequals the last digit of your university student number multiplied by 100 (Nomor Induk Universitas/NIU)and the angular frequency w equals the last digit of your student number (Nomor Induk Universitas/NIU).m) Modify your answer in point g to account for the external force F and write down the non-homogeneousdifferential equation!n) Find the complementary solution x(1) to the differential equation in point m! Use the mass of the firstpassenger, m=100 kg. (Hint: complementary solution refers to the solution of the homogeneous versionof differential equation in point m!)o) Find the particular solution x,(1) to the differential equation in point m!p) From your answer in point n and o, find the position of the passenger x(t) if the passenger was initially atx(0) = 0 moving to the left with speed 10 m/s!q) Now, if the angular frequency of the external force w is modified to w =prove that the particularsolution x,(t) to the non-homogeneous differential equation will beXp(t) = Aot sin (t).Find the value of Ao in terms of A, k, and m! Calculate the numerical value of Ao!r) Will the movement of the passenger stabilize over time (x(t) goes to zero)? If not, is it possible to make itstable by modifying the spring constant k? Explain your answer! 2. [LO2] Vehicle Suspension SystemSpring constant kPassengerwith massFix BaseтDamping constant cConsider a safety suspension system designed to protect passengers from an impact in the event of a vehicleaccident as shown in the above figure. The suspension system can be modelled as a spring-damper system withspring coefficient k and damping constant c. The passenger can be modelled as a point-mass with mass m andwe assume that he/she uses a seatbelt in such a way that his/her body is always connected to the suspensionsystem. We will use the concept of differential equation to predict the behaviour of this suspension systemunder various conditions.Case 1: We assume that the passenger is passive and apply no external force to the system.a) Use Newton's Second Law to write down the passenger's equation of motion in the form of a second orderdifferential equation as followsd²x+ az + azx = 0.dxdt²Write down the values of a,, az, and a; in terms of m, k, and c!b) Write down the characteristic polynomial of the differential equation in point a!c) Suppose that c=400 kg/s, k=500 N/m. Find the roots of the characteristic equation for 3 differentpassengers with mass m=100 kg, m=60 kg, and m=80 kg!d) Write down the position of the previous 3 different passengers x(t) if the passenger was initially at x(0) =0 m, moving to the left with speed 10 m/s! Will the movement of the passenger stabilize over time in theprevious 3 cases?e) The first passenger (with mass m=100 kg) complains that his/her suspension system oscillates too much.He/she wants you to modify his/her suspension to behave similarly to the behaviour experienced bypassenger 2 (with mass m=60 kg). Can you modify the spring constant k and the damping constant c tofulfil the request of passenger 1?f) Is it possible that for a certain passenger with mass M, the suspension system will fail to suppress themovement such that the passenger keeps getting faster? Explain your answer!

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Accepted Answer

Parts, g,h, i have been answered. Next three sub questions are solved in this solution