EXAM I, PHYSICS 4304 March 2, 2004 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 ALL work, 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 will count 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 THE ABOVE SIMPLE INSTRUCTIONS!!! FAILURE TO FOLLOW THEM MAY RESULT IN A LOWER GRADE!! THANK YOU!! NOTE!!!! Work any four (4) of the six problems. Each problem is equally weighted and worth 25 points for a total of 100 points on this exam. 1. A mass m is moving on a horizontal surface where the retarding force is proportional to the ½ (one-half) power (square root) of the velocity: F = -mkv½, where k is a constant. At t = 0, the velocity is v0 and m 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 position as a function of velocity (x(v)). (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) 2. A mass m moves along the x-axis. At time t = t0, its velocity is v0 and it is at x = 0. At t = t0, a time-dependent force given by F = F0(t0/t) begins to act, where F0 is a constant. Find: a. The velocity as a function of time (v(t)). (9 points) b. The position as a function of time (x(t)). (9 points) c. By making the appropriate Taylor’s series expansions of the results of parts a. and b., find expressions for v(t) and x(t) when (t - t0) << t0. In a few complete and grammatically correct English sentences, explain what this small time means physically. Do v(t) and x(t) resemble any familiar results in this limit? (7 points) The following integrals might be useful. The constants of integration are not shown. The lower limits are important in this problem! ∫t-1 dt = ln(t), ∫ln(at)dt = t[ln(at) –1] (a is any constant). The following Taylor’s series expansion might be useful: If z <<1, ln(1+z) z – (½)z2 + … NOTE!!!! Work any four (4) of the six problems. 3. 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 – (⅔)(x/a)3] (a & U0 are 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. 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. (4 points) d. If m is near the stable equilibrium point and has an energy E = (U0/6), set up the equation to solve for the turning points. Don’t try to solve it! (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).] (5 points) 4. Consider a one-dimensional periodic force, F(t). Over one time period τ = (2π/ω), it has the form F(t) = F0(t/τ), - (π/ω) < 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 displacement xs(t) of a damped oscillator with a sinusoidal driving force, along with the results of part a), to express the 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. 5. 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 which contains 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 a & b, assume that this motion is simple harmonic. a. Find the frequency of these oscillations. Compute a numerical value for this frequency for L = 0.2 m, A = 0.1 m2, and ρ = 2,000 kg/m3. (8 points) b. Derive an expression for z(t) that satisfies the initial conditions described in the problem statement. Let z0 = 0.01 m. (4 points) c. 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 = 2 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) d. 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) 6. A mass m = 5 kg is attached to an ideal spring of constant k = 10 N/m. At time t = 0 it is released from the origin (x = 0) with initial speed v0 = 12 m/s. Consider this oscillator under the following three DIFFERENT conditions: a. WITH NO FRICTION. Compute numerical values for the frequency, the period, the maximum velocity, the maximum amplitude, the maximum acceleration, and the total mechanical energy. Using (some of) these values, write an expression for the time dependence of the displacement x(t) (that is, use the computed information to evaluate the integration constants in the general solution for x(t).) (8 points) b. WITH FRICTION. A damping force proportional to the velocity is present. The damping force constant is b = 3 N·s/m. Compute numerical values for the frequency, the period, and the decrement of the motion. Calculate the time it takes for the amplitude to reduce to half of its maximum value. Using (some of) these values, write an expression for the time dependence of the displacement x(t) (that is, use the computed information to evaluate the integration constants in the general solution for x(t).) Is this oscillator underdamped, overdamped, or critically damped? (8 points) c. DRIVEN. Consider the damped oscillator of part b. A sinusoidal driving force F(t) = F0cos(ωt) is applied. Assume that enough time has gone by for the oscillator to reach a steady state. When numbers are needed, let F0 = 2 N. Under these steady state conditions, compute numerical values for the resonance frequency, the phase angle at the resonance frequency, and the maximum amplitude at the resonance frequency. (9 points)

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# EXAM I, PHYSICS 4304 March 2, 2004 Dr. Charles W. Myles

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