Physics 105, Spring 2015, Reinsch Homework Assignment 1 (This is a five page document. More hints added Feb. 2, 9pm. Problem statements unchanged.) Due Wednesday, February 4, 5:00 pm Problem 1 A particle moves along the x-axis in a potential U(x) = C 1 x 2 + C 2 x, where C 1, C 2 and x are greater than zero. (a) Find the equilibrium location. (b) If the mass of the particle is m, what is the frequency ω of small oscillations about the equilibrium location? (c) Let A denote the amplitude of oscillation. For the very small values of A considered in part (b), the frequency did not depend on A. As we start to consider larger values of A, the frequency starts to change slightly depending on the value of A. Compute the first correction to the result in part (b). This notion of small but not very small is explained in problem 4.39 (c). Problem 2 We consider a weakly damped oscillator without a driving force, as discussed in Chapter 5. The initial value of the velocity is zero, and the initial value of x is A. Using power equals force times velocity, and considering the time interval from zero to infinity, set up integrals and solve them to find (a) the total work done by the spring on the particle. (b) the total work done by the frictional force on the particle. This should be negative. (c) How doyour answers to parts (a) and (b) relate to the initial and final values of thespringpotential energy and the initial and final kinetic energies of the particle? Problem 3 In Eq. (5.68), (a) which symbols depend on the initial conditions, x(0) and ẋ(0)? (b) under what conditions will δ equal δ tr? What do the solution sets look like in the space whose axes are x(0) and ẋ(0)? Problem 4 Derive a formula similar to Eq. (6.41) for an oblate spheroid. Let R p be the distance from the center to the North or South Pole. Let R e be the distance from the center to the equator. Look up the definitions of geodetic latitude and geocentric latitude. In this problem we will use the usual geocentric co-latitude, and call it θ.
Problem 5 A spherical planet is covered entirely with water except for two small islands, A and B. The longitude of A is φ o, and the longitude of B is φ o (in the interest of fairness, the Prime Meridian cannot go through either island). The co-latitude of A is θ A, and the co-latitude of B is θ B. The speed limit for boats depends on which hemisphere you are in. On the A side of the Prime Meridian, the speed limit is v A. On the B side of the Prime Meridian, the speed limit is v B. We are interested in the fastest path from A to B. Let θ o be the value of θ at the point where the path crosses the Prime Meridian. (a) If v A = v B, what is the fastest path? You may use the result given in Problem 6.16. What is the value of θ o? (b) If v A = v o + v and v B = v o v, where v is small, what is the first nonvanishing correction to θ o?
Hints for Problem 1 (c) Begin by sketching the function U(x) for both positive and negative x. Although the particle s motion is constrained to x > 0, we will need to understand the behavior of the three solutions to a certain cubic equation. Let s call the location of the equilibrium x eq. For a value of E greater than U(x eq ) draw a horizontal line in the diagram and note that there are three intersections with the U(x) curve. Let s call these x N, x L (for negative, left and right ). The turning points are x L. The location of the equilibrium lies approximately midway between them. We could write out formulas for these roots, but the solutions to cubic equations are messy and not illuminating. We should understand however that x R, for example, is a function x R (E,C 1,C 2 ). We ll use logic similar to that in Eq. (4.57) and Eq. (4.58) in the text. Show that E U(x) is a constant times (x R x)(x x L )(x x N )/x. (You may use the fact that if two cubic polynomials have the same roots, and if the roots are distinct, then one polynomial is a multiple of the other.) Using this form, the integral for the period can be done exactly. It is a complete elliptic integral of the third kind. At this point, we have an exact formula for the period as a function of E with implicit functions such as x R (E,C 1,C 2 ). As E approaches U(x eq ), the values of x N, x L approach limits that are easy to evaluate (no messy formulas). For E slightly greater than U(x eq ), we can write approximate formulas for x R (E,C 1,C 2 ), etc. For x N this can be done with a linear correction to the value computed above. For x L we begin with an analysis based on a second-order Taylor expansion of U(x) about x eq. This results in x R and x L being x eq ± A 1, where A 1 is defined to be the amplitude in this quadratic well. However, this approximation for x L is not good enough. We need to go to the next order in A 1. This can be done by computing the slope of the parabola at the two first-order locations found, and then adjusting their positions according to the value of the cubic term at those locations. As discussed above, the maximal excursions to the right and left of the equilibrium point are slightly different, so the notion of an amplitude is not defined in its usual sense. If we use (x R x L )/2 as the definition of A then this is actually equal to A 1 to the order that we are working. Thus in the final result A 1 will be replaced with A. Using (x R x eq ) as the definition of A would result in changes in terms of higher order than the one we are computing. It is in this sense that we can talk about the amplitude A. At this point students can either use the complete elliptic integral of the third kind and use software such as Mathematica to compute its derivatives, which will be necessary to compute T to second order in A 1, or students can proceed with several simpler integrals. This can be done by defining x f(x) = (1) x xn In the integral for T, f(x) is then replaced with its second-order Taylor series about x eq. The resulting integral can then be solved using standard results, such as xr x L dx (xr x)(x x L ) = π (2)
Finally, we use ω = 2π/T to compute ω to second order in A 1, which is the same as second order in A as discussed above. Here is a numerical example for the integral x R x L for C 1 = 1J/m 2,C 2 = 2Jm,E = 3.0005J. xdx (xr x)(x x L )(x x N ) x_eq [m] 1.000000000000000 U(x_eq) [J] 3.000000000000000 exact roots [m] -2.000111109053536, 0.9871457295676324, 1.012965379485904 approx roots [m] -2.000111111111111, 0.9871456110681975, 1.012965500042914 exact integral 1.813774174762992 approx integral 1(c) 1.813774172576381 approx integral 1(b) 1.813799364234218 Comment on Problem 2 The symbol A is used in the problem statement. Unfortunately this symbol is used in the text for something else. In Eq. (5.38) it plays a different role. Let s use the symbol A text for the A in the text. Thus, Eq. (5.38) for t = 0 becomes x(0) = A text cos(δ). This ability to adapt the notation will be important in your future careers when you encounter problems in settings where the notation is different than that in the textbook. Hints for Problem 3 Set up the equations that determine A tr and δ tr. Solve for A tr cosδ tr and A tr sinδ tr. Based on this, show that the unit vector (cosδ tr,sinδ tr ) points in the same direction as a certain vector that depends on the initial conditions. State what the solution sets look like in the space of initial conditions (the space whose axes are x(0) and ẋ(0)) in general terms, using words like straight line, circle, ellipse, hyperbola, or ray. You do not have to provide detailed formulas or statements about where the curve(s) cross the axes, etc. Hints for Problem 4 Note: this problem would be much easier if we were using a different definition of θ. The reason we are using the geocentric co-latitude is to get some practice computing the various derivatives that occur and also to make the problem different from well known published results about oblate spheroids. Write out the equation for the oblate spheroid in cylindrical coordinates using ρ and z. Work out relationships between dz, dρ and dθ Based on this, write out a formula for ds 2.
A slight re-arrangement then gives the integral we after. Note we are not using the Euler-Lagrange Equations in this problem. Hints for Problem 5 If you know of two linearly independent vectors in a plane, the plane can be described as the set of vectors orthogonal to the cross product of the two vectors. Use cross products and dot products of unit vectors. There is a simple way to find the angle between two unit vectors from their dot product. Note we are not using the Euler-Lagrange Equations in this problem. This problem is similar to Prob. 6.4 in the text in that regard.