EXAM I, PHYSICS 4304

October 24, 2005

Dr. Charles W. Myles

INSTRUCTIONS: Please read ALL of these before doing anything else!!!

  1. PLEASE write on one side of the paper only!! It wastes paper, but it makes my grading easier!
  2. PLEASE don’t write on the exam sheets, there won’t be room! If you don’t have paper, I’ll give you some.
  3. PLEASE show ALLwork, writing down at least the essential steps in the problem solution. Partial credit will be liberal, provided that essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work.
  4. The setup (PHYSICS) of a problem counts more heavily than the detailed mathematics of working it out.
  5. PLEASE write neatly. Before handing in your solutions, PLEASE:a) number the pages and put them in numerical order, b) put the problem solutions in numerical order, and c) clearly mark your final answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves.

NOTE: I HAVE 20 EXAMS TO GRADE!!! PLEASE HELP ME GRADE THEM EFFICIENTLY BY FOLLOWING THESE SIMPLE INSTRUCTIONS!!! FAILURE TO FOLLOW THEM

MAY RESULT IN A LOWER GRADE!!

THANK YOU!!

NOTE!!!! Work four (4) of the six problems. Each is equally weighted & worth 25 points for a total of 100 points.

  1. A mass m moves on a horizontal surface where the retarding force is proportional to the ¼(one-fourth) power (fourth root) of the velocity: F = -mkv¼, where k is a positive constant. At t = 0, the velocity is v0m is at x = 0. Find (in any order):

a.The velocity as a function of time (v(t)). (5 points)

b.The time it takes the mass to stop. (5 points)

c.The velocity as a function of position (v(x)). (5 points)

d.The position as a function of time (x(t)). (5 points)

e.The distance the mass travels before it stops. (5 points)

The following integral might be useful. The constant of integration is not shown. The lower limit is important in this problem! ∫vp dv = (vp+1)/(p+1), where p is any power (p  -1)

  1. Consider a one-dimensional periodic force, F(t). Over one time period τ = (2π/ω) (ω is a frequency) it has the form: F(t) = - F0 ωt, - (π/ω)  t < 0

= F0 ωt, 0 < t  (π/ω), where F0 is a constant.

a.Find the Fourier series representation of F(t). (15 points)

b.Consider a one-dimensional, damped, driven harmonic oscillator with a driving force given by F(t), above. Use the known solution for the steady state displacementxs(t) of a damped oscillator with a sinusoidal driving force, along with the results of part a), to find the steady state solution for x(t) for this driven oscillator as a series solution. (10 points)

The following integrals might be useful. The constants of integration are not shown.

The lower limits are important in this problem!

∫x cos(ax)dx = (a-2)cos(ax) + (xa-1)sin(ax) ∫x sin(ax)dx = (a-2)sin(ax) - (xa-1)cos(ax)

NOTE!!!! Work any four (4) of the six problems.

  1. A mass m moves onthe x-axis. At time t = 0 its velocity is v0 it is at x = 0. At t = 0, a time-dependent force given by F = F0e-kt begins to act, where F0k are positive constants. Find:

a.The velocity as a function of time (v(t)). (5 points)

b.The position as a function of time (x(t)). (5 points)

c.Consider v(t)x(t) in the limit of very small (but not zero!) time. Make the appropriate Taylor’s series expansions of the results of parts a.b., and find expressions for v(t)x(t) when kt < 1. In complete, grammatically correct English sentences, briefly explain what this small time limit means physically. Do v(t)x(t) resemble any familiar results in this limit? If so, tell me what these are. (5 points)

d.Consider v(t)x(t) for large times. Take the limit of results of parts a.b. for very large times (kt > 1). the In a complete, grammatically correct English sentences, explain what this long time means physically. Do v(t)x(t) resemble any familiar results in this limit? If so, tell me what these are. (5 points)

e.When first formulating this problem, I originally was going to ask you to compute the time it takes the mass to stop. However, I then realized that computing this in the usual way (finding the time t when v(t) = 0) leads to unphysical results. Despite this, compute this time by this method anyway. In a complete, grammatically correct English sentence, explain why this is unphysical. (5 points)

The following integral might be useful. The constant of integration is not shown. The lower limit is important in this problem!∫e-kt dt = (e-kt)/(-k). The following Taylor’s series expansion might be useful: For |kt| <1, e-kt  1-kt + (½)(kt)2 - …

  1. A mass m is confined to the x-axis by a conservative force which is derivable from a potential energy function given by U(x) = U0[(x/a)2 –(x/a)4] (a U0are positive constants).

a.Compute the force F(x). (3 points)

b.Find the equilibrium points. Determine whether these are points of stable or unstable equilibrium. Also find the values of x where U(x) = 0. Use these results to SKETCH the potential U(x). (5 points)

c.Assume that m has an energy very near the stable equilibrium point, so that it doesn’t move very far away from that point. Compute the angular frequency ω of oscillations about that point. (5 points)

d.If m is near the stable equilibrium point and has an energy E = U0, set up and solve the equation to solve for the turning points. (Hint: You should get a quadratic equation in x4, instead of x2, which can be solved with the quadratic formula.) (4 points)

e.If m is moving in the +x-direction, compute the minimum velocity it must have at x = 0 to escape to infinity. (4 points)

f.At t = 0, the mass is at x = 0 and it is moving in +x direction with a velocity equal to that found in part e. Go through the steps necessary to calculate the time as a function of position, t(x). You’ll get a messy integral that isn’t easily done in closed form. Leave it in integral form! [If this integral could be done in closed form, t(x) could, in principle, be inverted to obtain the position as a function of time, x(t).] (4 points)

NOTE!!!! Work any four (4) of the six problems.

  1. Parts c.is (obviously!) independent of parts a.b.
  1. Amass mis confined to the x-axis by a conservative force derivable from a potential similar to that of Problem 4: U(x) = kx2 - εx4.Make a sketch of the velocity versus position phase diagram for this nonlinear oscillator for the following total mechanical energies: E = (k2/ε), E = (k2/ε). (8 points)
  2. Assume that the termεx4 is much smaller than kx2 so that this is almost a harmonic oscillator. Assume initial conditions that at t = 0, m is at rest at x = A. Use the method of successive approximations (as follows) to find an approximate solution, x(t), to the nonlinear equation obtained in c. As an initial approximation, take the harmonic oscillator solution (solution when εx4 = 0)for x(t)which satisfies these initial conditions. Use this approximation in the equation of motion to obtain the next approximate solution. Make sure that this new solution satisfies the initial conditions! Assume that ε is small enough that this process can be stopped after one iteration. How does this non-linear solution differ from that for the simple harmonic oscillator? If you can, discuss this physically, as well as mathematically. (8 points)
  3. By a successive approximation procedure, use your calculator to find x in the equation x = e-x. Obtain a result accurate to 4 significant figures. Use either direct iteration or Newton’s Method. I recommend the latter! Hint: Make an initial “guess” in the range 0 < x. Before choosing this initial guess, it might be useful to make a sketch of xand e-xon the same graph to give you an idea where the two functions cross! (9 points)
  1. See figure.A liquid of mass density ρ is placed in a U-tube of cross sectional area A. The length of the portion of the tube containing the liquid is L. The liquid is initially in equilibrium. It is then displaced a small distance z0 from equilibrium and released from rest. It then “sloshes” or oscillates up and down (the distance z in the figure changes). For parts ab, assume that this motion is simple harmonic.
  1. Find the frequency of these oscillations. Compute a numerical value for this frequency for L = 0.18 m, A = 0.11 m2, and ρ = 2,100 kg/m3. (8 points)
  2. Derive an expression for z(t) that satisfies the initial conditions described in the problem statement. Let z0 = 0.011 m. (4 points)
  3. Suppose now that the liquid has viscosity η, so that the oscillations will be damped. Assume that the retarding or damping force is proportional to the velocity and has the form Fr = - η(A/L)(dz/dt). Assuming that this motion corresponds to that of a damped oscillator, derive an expression for the damping constant β. If η = 3.5 10 –3 N·s/m2, compute a numerical value for β. (Use also the numerical values for L, A, and ρ given in part a.) (8 points)
  4. For the damped oscillator motion of part c., write an expression for z(t) that satisfies the initial conditions described in the problem statement. Is this oscillator underdamped, over damped, or critically damped? (5 points)