4.4. A spring-mass system is subjected to a harmonic force which has an amplitude 30N and frequency 20 rad/s. The system has mass m = 5 kg, and stiffness coefficient k = 2 x10° N/m. The initial conditions are such that xo = 0, io 2 m/s. Determine the displacement, velocity, and acceleration of the mass after 0.5, 1, 1.5 s.

Consider the double mass/double spring system shown below. - click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F friction We can write the resulting system of second-order DEs as a first-order system, w'(t) = Aw(t), with w = (x₁, x₁, x₂, x₂) T For values of k = 4, m = 1 and c = 21,2 = 0.5±3.2i, V₁ 23,4= - 0.5 ± 1.13i, V3 1, the resulting eigenvalues and eigenvectors of A are -0.039 0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i' 0.409 -0.216 - 0.489i 0.661 and (a) Find a set of initial displacements x₁(0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. (x₁(0), x₂(0)) Enter your answer using angle braces, (and ). cx (friction proportional to velocity). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

6. A 1kg mass is attached to a spring (with spring constant k = 4 N/m), and the spring itself is attached to the ceiling. If you pull the mass down to stretch the spring past its equilibrium position, when you release the mass and observe its (vertical) position, it's said to undergo simple harmonic motion. AT REST MASS PULLED DOWN wwww Under certain initial conditions, the mass's vertical position (in metres) relative to its equilibrium position at time t, y(t), can be modelled by the equation y(t) = cos(2t) – sin(2t), (Note that y measures how much the spring has been stretched, so y = 1 indicates the mass is Im below its equilibrium position, whereas y = -1 indicates it is 1m above its equilibrium position.) (a) Find expressions for the mass's (vertical) velocity v(t) (relative to its equilibrium position) and the mass's (vertical) acceleration a(t) (relative to its equilibrium position). (b) Is the mass moving toward the ceiling or toward the floor at t= T? Justify your answer with…

A 2-kg block attached to an Ideal massless spring and fixed on the other end on a horizontal frictionless surface. This block oscillates with an amplitude of 0.5 m and reaches a maximum speed of 2 m/s. What is the spring constant in N/m?