Activity Based Physics Thinking Problems in Oscillations and Waves: 
1) A mass is attached to two heavy walls by two springs as shown in the figure below. Assume the mass slides on a frictionless floor and neglect damping effects in the springs. When positioned as shown, each of the springs is at its rest length and they have the spring constants indicated.
(a) Write the equation of motion for the mass. 2) Observation of a mass on the end of a spring reveals that the detailed structure of the position as a function of time is fit very well by a function of the form x(t) = A cos(wt + f). Yet subsequent observations give convincing evidence that this cannot be a good representation of the motion for long time periods. Explain what observation leads to this conclusion and resolve the apparent contradiction.
(a) If the block starts at time t=0 with the spring being at its rest
length but the block having a velocity v_{0}, find a solution for
the mass's position at all subsequent times. Make any assumptions you like
in order to have a plausible but solvable model, but state your assumptions
explicitly.
kinetic, potential, or a mixture of the two? (This is not a short answer question. Show how you know.) 4) A class looked at the oscillation of a mass on a spring. They observed for 10 seconds and found its oscillation was well fit by assuming the mass's motion was governed by Newton's second law with the staticspring force where s represents the stretch of the spring. However, it was also clear that this was not an adequate representation for times on the order of 10 minutes, since by that time the mass had stopped oscillating. (a) Suppose the mass was started at an initial position x_{0}
with a velocity 0. Write down the solution, x(t) and v(t), for the equations
of motion of the mass using only the staticspring force. What is the total
energy of the oscillating mass?
7) Solving the equation of motion of a mass hanging from a spring, we obtain the solution x[t] = A cos wt. If an actual mass is hung from a spring and data is taken using a sonic ranger, two problems are observed: the displacement curve does not start at its maximum value, and the oscillation diminishes over time. Discuss what was the difficulty with our mathematical model of the system and how you would fix the two problems. 8) As a lecture demonstration, a Professor starts a mass on a spring oscillating. After a few minutes, the oscillation died down. The mass used had a mass of M = 50 grams, the spring had a spring constant of k = 5 Newtons/meter and the spring had a mass of m = 5 grams. Estimate the rise in temperature of the spring. State clearly all the numbers you are estimating and begin with a few sentences explaining what physical principles will be relevant in carrying out your calculation. 9) A 50 gram mass is hanging from a spring whose unstretched length is 10 cm and whose spring constant is 2.5 N/m. In the list below are described five situations. In some of the situations, the mass is at rest and remains at rest. In other situations, at the instant described, the mass is in the middle of an oscillation initiated by a person pulling the mass downward 5 cm from its equilibrium position and releasing it. Ignore both air resistance and internal damping in the spring. At the time the situation occurs, indicate whether the force vector requested points up (U), down (D), or has magnitude zero (0). (a) The force on the mass exerted by the spring when the mass is at
its equilibrium position and is at rest. 10) In a nucleus that undergoes nuclear fission, the parts of the nucleus repel each other because they all have the same electric charge, but attract each other because of the strong nuclear force. Let's start on building up a model of a nucleus that consists of two (reasonably) small particles of mass m, that each have a charge q, and that are connected by a spring of spring constant k. Let's also assume that the spring has a rest length of 0.
(a) What will be the equilibrium separation, L, between the two
charges? Express your answer in terms of the given parameters of the system.
12) The following problem is found in Halliday and Resnick's Fundamentals of Physics, a popular introductory text. A block of mass M at rest on a horizontal frictionless table. It is attached to a rigid support by a spring of constant k. A bullet of mass m and velocity v strikes the block as shown in the figure. on the next page. Determine (a) the velocity of the block immediately after the collision and (b) the amplitude of the resulting simple harmonic motion.
In order for you to solve this problem you must make a number of simplifying assumptions, some of which are stated in the problem and some of which are not. First, solve the problem as stated. Then (c) discuss the approximations which you had to make in order to solve the problem. I can think of at least five.
14) A mass m on a spring with spring constant k is started at time t = 0 with a velocity v_{0} at position x_{0} = 0. (a) Assuming no damping, what is the equation of motion the position
of the mass satisfies?
16) A mass is hanging from a spring off the edge of a table. The position of the mass is measured by a sonic ranger sitting on the floor 25 cm below the mass's equilibrium position. At some time, the mass is started oscillating. At a later time, the sonic ranger begins to take data. Below are shown a series of graphs associated with the motion of the mass and a series of physical quantities. The graph labeled (A) is a graph of the mass's position as measured by the ranger. For each physical quantity identify which graph could represent that quantity for this situation. If none are possible, answer N. a) velocity of the mass b) net force on the mass

* To download an executable of the program HARMOSC, click on the program name. This program is a DOS program. (It can be run from Windows, but you will probably need to create a PIF file to do so.) To run it you must also have a "Borland Graphics Interface" (BGI) file appropriate for your graphics screen in the same directory as the program. For most computers today, the appropriate file is "EGAVGA.BGI". Both the program and the BGI file are contained in a "zip" file. Unzip them into the same directory using PKUNZIP or WINUNZIP.
These problems written and collected by E. F. Redish. These problems may be freely used in classrooms. They may be copied and cited in published work if the Activity Based Physics Thinking Problems in Physics site is mentioned and the URL given. Web page created and edited by K. A. Vick.
To contribute problems to this site, send them to redish@physics.umd.edu.
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University of Maryland PERG
Comments and questions may be directed to
E. F. Redish
Last modified June 21, 2002