A 64 pound weight is attached to the end of the spring.  After reaching the equilibrium position the spring is stretched one foot below the equilibrium position.  The weight is then released and as it is released it is struck a downward blow giving it an initial velocity of 2 ft/sec.  Take the moment the weight is released and struck as time zero.  At time zero a periodic external force given by F(t) = (1/2)cos(4t) pounds begins acting on the system.  t is time in seconds.  Consider the damping factor to be negligible, i.e., take k₂ to be zero.  The spring constant is 32.  Find the function giving y, the position of the bottom of the weight as a function of time given in seconds.  Find y after four seconds.  Find y after one hour.  Find y after one day based on the mathematical model.  What is going to happen to the spring?  The graph below is an attempt by computer software to show y as a function of time over the first 600 seconds.  The graph may be misleading since the function is periodic and would go through almost 382 periods in 600 seconds.  The graph does give a sense of what is happening to the amplitude.  Look at the second graph below for more accuracy but a shorter time interval (about 20 periods).  Use it to check your answer.  The graph on the right below shows one fundamental period of the position function.

Find the function that gives the velocity of the object attached to the end of the spring.  What is wrong with the mathematical model for large t?  You may be helped looking at this animation of the action of the spring and weight as t goes from 0 to 80pi seconds (Quicktime Version).  In the animation the spring is 10 feet long in the equilibrium position and the cross section of the weight is a circle with a diameter of 2 feet.  When t reaches 80pi in the animation, the animation repeats itself starting again from t = 0.  The picture at the right shows the position of the spring and weight at time t = 0 seconds.

Resonance in Three Dimensions:

Tacoma Narrows Bridge Collapse

Long Version

Wind Version

Animation Corrected by Bryan Paul

Solutions at the bottom of the page

Extra Credit:  The vibrating spring with constant velocity over intervals

Find the governing differential equation and position functions for a 32 pound object attached to the end of a spring with a spring constant of 1 and a forcing function that yields a constant velocity in the direction of motion.  This velocity changes sign periodically.  The forcing function is piecewise.  The object is pulled down until the spring is stretched to 5 feet below its equilibrium position and then the object is released with an initial velocity of -1 ft/sec and the forcing function produces a constant velocity of -1 ft/sec.  After the object has traveled 10 feet it is impeded and reverses direction with an initial velocity at that point of 1 ft/sec. and the forcing function changes to produce a constant velocity of 1 ft/sec.  This behavior continues indefinitely.  Click here to see an animation of the motion with no scales and click here to see the animation with scales on the y-axis.  Take y = 0 to be the equilibrium position of the bottom of the object and y positive to indicate below the equilibrium position (contrary to the labeling in the animation).  Neglect damping.

You may replace one problem on the in-class portion of Exam II with the following initial value problem.  Check your solution by comparing its graph to the graph of the solution function given on the right.