A horizontal spring mass system oscillates on a frictionless plane.  At time t=0, it is moving left at position x=9 cm.  It has velocity v=0, at positions x=0 cm and 12 cm, and completes one full cycle in 2 seconds.  Write the position and velocity kinematic equations for this oscillating system, including the phase constant.

Problem 2:  Special sections of roadway are sometimes paved with “rumble strips” to alert inattentive drivers. In a particular case the grooves are spaced L = 0.26 m apart and the depth of each groove is d = 0.35 cm. As you drive over this rumble strip, the tires of your car oscillate about their equilibrium positions with a frequency of f = 68.6Hz. Refer to the figure, which is not drawn to scale.  Part (b) Find the vertical position of the tire, in centimeters, at the time t = 1.85 s. Part (c) With your tire oscillating at a frequency of f = 68.6 Hz and the distance between grooves L = 0.26 m, what is the speed of your car, in kilometers per hour?

A mass-spring motion is governed by the ordinary differential equation d²x dx +b + k(t)x= F(t), dt² dt m where m is the mass, b is the damping constant, k is the spring constant, and F(t) is the external force. We consider the initial conditions x(0) = 1 and x/(0) = 0. Assume the following numerical values for this part of the project: m = 1, k = 1/5, b= 1/5, and F(t) = cos yt. 1 (a) Read section 4.10. Explain what is the resonance frequency, and then compute the resonance frequency for this mass-spring system. (b) The ODE45-solver can be used to obtain the solution curves in MATLAB. Use the script Project2_Q2.m to plot the solutions and estimate the amplitude A of the steady response for y = 0.2, 0.42, 0.6, and 0.8. (c) The script also provide you with the graph of A versus y. For what frequency 7 is the amplitude the greatest? Is it equal to that you obtained in (a)?

Two particles oscillate in simple harmonic motion with amplitude A about the centre of a common straight line of length 2A. Each particle has a period of 1.5 s, and their phase constants differ by 4 rad. (a) How far apart are the particles (in terms of A) 0.5 s after the lagging particle leaves one end of the path? Enter the exact answer in terms of A. ab sin (a) Ω