5. ( lowing IVP, A particular mass-spring system can be described using the fol- u" + 4 u = sin(t), u(0) = 0, u'(0) = 0. %3| Find the appropriate form of the displacement u(t) at any time t, and ex- plain the behavior of the system whem t → o. Sketch the graph of the displacement function u(t). At which time t1 (t1 > 0) the displacement u(t) would be zero?

5. ( lowing IVP, A particular mass-spring system can be described using the fol- u" + 4 u = sin(t), u(0) = 0, u'(0) = 0. %3| Find the appropriate form of the displacement u(t) at any time t, and ex- plain the behavior of the system whem t → o. Sketch the graph of the displacement function u(t). At which time t1 (t1 > 0) the displacement u(t) would be zero?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Identify the Cauchy-Euler equation. d*y ,3 dy 52 dz + 8y = 0 - dz4 O 2a° y" – 7xy' + 6y = 0 O a*y" + 2x?y" – y' + 5y = 0 d³y d²y dy 2x2. dr2 + 5x- + 3y = 0 dx dr3 « Previous a. hp
The first condition allows computation of the spring constant, k. It follows that the damped spring-mass system described in this problem satisfies: y" +2y'+ 10y = 0.25 sin(4t). with y(0) = y'(0) =0 Apply the method of Undetermined Coefficients
B4.
2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = = x²ex. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of vand x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mv = F) (iii) Assume the particle starts in x0 = 0 with positive initial velocity vo > 0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ô, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 ༧(2) vo m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > v. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the…
H2. Suppose a particle of constant mass m with position x > 0, moves in one space dimension under the influence of the gravitational force of another point particle of constant mass M sitting at x = 0, i.e. the attracting force is F (a) Using Newton's second law, show that d à (2m²² - dt GmM i. GmM x = 0. (i.e., the total energy, sum of kinetic and potential energy, is conserved). (b) Using the change of variables var = r, solve the equation of motion and determine the velocity of the particle v(x) as a function of x assuming it starts with zero velocity at xo. Does the particle's speed in x = 0 depend on the initial position?
5. Consider the undamped spring equation y" + cy= sin(2t). (a) For what value of c does resonance occur? Compute the solution at resonance with y(0) = 1 and y'(0) = 0. (b) For what values of c is there a beat with frequency 0.1 Hz? (The beat frequency is defined as rad/s where is the natural frequency of the spring and w is the forcing frequency.) ۳-w 2 1
Given a rectangle with side a = 20 cm, find the length (a + b) and b if the Golden Ration is to be preserved?
Please I need help thank you