This problem deals with a mass in on a spring (with constant k) that receives an impulse
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- As we showed, the horizontal displacements ξi of masses i = 1 . . . N satisfy equations of mo- tion md2ξ1 =k(ξ2−ξ1)+F1, dt2 md2ξi =k(ξi+1−ξi)+k(ξi−1−ξi)+Fi, dt2 md2ξN =k(ξN−1−ξN)+FN. dt2 where m is the mass, k is the spring constant, and Fi is the external force on mass i. In Exam- ple 6.2 we showed how these equations could be solved by guessing a form for the solution and using a matrix method. Here we'll solve them more directly. a) Write a python program to solve for the motion of the masses using the fourth-order Runge- Kutta method for the case we studied previously where m = 1 and k = 6, and the driving forces are all zero except for F1 = cos ωt with ω = 2. Plot your solutions for the displace- mentsξi ofallthemassesasafunctionoftimefromt=0tot=20onthesameplot. Write your program to work with general N, but test it out for small values—N = 5 is a reasonable choice. You will need first of all to convert the N second-order equations of motion into 2N first- order equations.…arrow_forwardConsider the Z-transform of the impulse response h(n) shown below. If we define y(n) as the output of a relaxed system from the system described by its impulse response h(n), and input signal, x(n), determine the following with included systematic and legible solutions:a. Output response from an impulse inputb. Output response from a unit step inputarrow_forwardConsider a system with input x(t) and output y(t) . The relationship between input and output is y(t) = x(t)x(t − 2) a. Is the system causal or non-causal?b. Determine the output of system when input is Aδ(t) , where A is any real orcomplex number?arrow_forward
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- Electromagnetic Pulse propagating at oblique angle to a dielectric interface Consider a gaussian wave pulse propagating along the z-axis from region 1 with refractive index n1 and onto a dielectric interface y = m z (for all x). To the left of this dielectric interface, the refractive index is n2. Devise an initial value computer algorithm to determine the time evolution of the reflected and transmitted electromagnetic fields for this pulse. e.g., n1 = 1 , n2 = 2 initial profile (t = 0, with z0 < 0) Ex = E0 exp[-a (z-z0)^2] By = n1 * Ex Choose parameters so that the pulse width is at least a fact of 8 less than the z- domain of integration ( -L < z < L). For the slope of the interface, one could choose m = 1.arrow_forwardSolve in PYTHON OR JUPYTER Consider the differential equation dy/dt = y(a −y^2) for the values a = −1, a = 0 and a = 1 and determine their critical points. Sketch for each of these differential equations their direction field and phase lines. Use these plots to determine whether the critical points are asymptotically stable or unstable.arrow_forwardHere is the Berkeley Madonna code: {Top model} {Reservoirs} d/dt (Q) = + Stimulus - Imemb INIT Q = -65/cap {Flows} Stimulus = Intensity*SquarePulse(3,.5) {at t=3 of 0.5 duration} Imemb = IL+IK+INa {Functions} Intensity = 100 {microamps} cap = 1 E = Q/cap {Submodel "INa_"} {Functions} ENa = 50 INa = gNa*(E-ENa) GNaMax = 120 gNa = GNaMax*m*m*m*h {Submodel "m_gates"} {Reservoirs} d/dt (m) = + m_prod - m_decay INIT m = am/(am+bm) {Flows} m_prod = am*(1-m) m_decay = bm*m {Functions} am = 0.1*(E+40)/(1-exp(-(E+40)/10)) bm = 4*exp(-(E+65)/18) {Submodel "h_gates"} {Reservoirs} d/dt (h) = + h_prod - h_decay INIT h = ah/(ah+bh) {Flows} h_prod = ah*(1-h) h_decay = bh*h {Functions} ah = 0.07*exp(-(E+65)/20) bh =…arrow_forward
- Simplify the following Boolean functions. F(w, x, y, z) = !w!xz + w!xz + !w!x!y!z + !w!xy!z + !wx!y!z + !wxy!z + wxy!z + wx!y!z + w!x!y!z + w!xy!z F(w, x, y, z) = _____answer_______ *Please do not include "F(w, x, y, z) = " in your solution. Add a single space between variables and operators.arrow_forwardChoose an elliptic curve and three points P,Q,R and construct via acomputer program (e.g. Geogebra) the two points (P ⊕ Q) ⊕ R andP ⊕ (Q ⊕ R). Drag the dots and explore what that means for associativity.arrow_forwardProvide a practical scenario to illustrate like the trolley problemarrow_forward
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