## Activity Based Physics Thinking Problems in Oscillations and Waves:Mass on a Spring

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### Activity Based Physics Thinking Problems in Oscillations and Waves: Mass on a Spring

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.
(b) Show that the equation of motion is the same as for a single harmonic oscillator but with a different spring constant. What will be the period of the mass's oscillation if it is displaced?

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.

 3) A block of mass m is attached to a spring of spring constant k which is attached to a wall as shown on the right.

(a) If the block starts at time t=0 with the spring being at its rest length but the block having a velocity v0, 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.
(b) Are the energies at the times

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 static-spring 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 x0 with a velocity 0. Write down the solution, x(t) and v(t), for the equations of motion of the mass using only the static-spring force. What is the total energy of the oscillating mass?
(b) Assume that there is also a velocity dependent force (the damping force) that the spring exerts on an object when it is moving. Let's make the simplest assumption, that the dynamic-spring force is linear in the velocity
Assume further that over one period of oscillation the velocity dependent piece is small. Therefore, take x(t) and v(t) to be given by the oscillation without damping. Calculate the work done by the damping force over one period in terms of the parameters k, m, and g and the fraction of the energy lost in one period.
(c) The mass used was 1 kg. From the description of the experiment, estimate the spring constant and the number of periods it took to lose half the energy. Use this and your result from (a) to estimate the approximate size of the damping constant g

 5) A mass, m, hangs from two identical springs with spring constant k which are attached to a heavy steel frame as shown in the figure on the right.  (a) If the system is at rest, what is the distance s0 that each spring is stretched?  (b) Suppose the mass is at a position which is a distance x above its equilibrium point. Identify all the forces acting on the mass and the net force acting on the mass. (Assume the mass remains doesn't tilt.)  (c) Write the equation of motion for the motion of the mass and show that it is the same as one would have if one hung the mass from a single spring of spring constant keff and find the value of keff.  (d) Write the solution to the equation of motion for the mass for the case that the mass is started at time t = t0 with a position x0 and a velocity 0.

 6) The graphs at right represent a computer simulation of a mass on a spring. The kinetic, potential, and total energy were plotted as well as the position and velocity of the oscillator. (Graphs generated by use of the program HARMOSC *.  The potential and kinetic energy curves oscillate, but not about zero. They also seem to oscillate twice as fast as the position and velocity curves. Is this correct? Explain.

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.
(b) The force on the mass exerted by the spring when the mass is at its equilibrium position and is moving downward.
(c) The net force on the mass when the mass is at its equilibrium position and is moving upward.
(d) The force on the mass exerted by the spring when it is at the top of its oscillation.
(e) The net force on the mass when it is at the top of its oscillation.

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.
(b) What will the total potential energy of the system be at equilibrium? Express your answer in terms of the given parameters of the system and, if necessary, universal constants.

 11) One end of a spring is attached to a wall, and the other to a cart as shown in the figure below. When the marker on the cart is at x=0 the spring is relaxed -- neither compressed nor stretched. The cart has a mass of 250 g, the rest length of the spring is 6 cm and it has a spring constant k = 0.5 N/cm. The cart is grabbed and pulled to the right a distance of 2 cm and held there.  (a) What forces are felt by the spring? Be sure to identify the type of force and the agent causing it in each case. What is the total force on the spring?  (b) What forces are felt by the cart? Be sure to identify the type of force and the agent causing it in each case. What is the total force on the cart?  (c) If the cart is released, with what period will it oscillate?

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.

 13) Two springs are joined and connected to a mass m as shown in the figure below. The surfaces are frictionless.  (a) If the force constants of the springs are k1 and k2, what is the frequency of the mass's oscillation? (Hint: Write the equation of motion for the mass after writing free body diagrams for each spring and for the mass.)  (b) What is the "effective spring constant" of the combination of k1 and k2? (That is, if the mass were hooked to a single spring, for what value of k would it have the same frequency as we have here?)

14) A mass m on a spring with spring constant k is started at time t = 0 with a velocity v0 at position x0 = 0.

(a) Assuming no damping, what is the equation of motion the position of the mass satisfies?
(b) Find an explicit general solution x(t) that satisfies this equation for any initial conditions.
(c) Find a particular solution that satisfies the initial conditions given. Demonstrate that for your solution x(0) = x0 = 0 and v(0) = v0
(d) For the particular case discussed here, what is the initial energy of the oscillator?
(e) Are the energies at the times  kinetic, potential, or a mixture of the two? (This is not a short answer question. Show how you know.)

 15) As part of a Physics Open House a Department sets up a bungee jump from the top of their physics building. Assume that one end of an elastic band will be firmly attached to the top of the building and the other to the waist of a courageous participant. The participant will step off the edge of the building to be slowed and brought back up by the elastic band before hitting the ground (we hope). Estimate the length and spring constant of the elastic you would recommend using. Picture Courtesy of Atti Laurila

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
c) force exerted by the spring on the mass d) kinetic energy of the mass
e) potential energy of the spring f) gravitational potential energy of 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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Comments and questions may be directed to E. F. Redish