PHYS 325 Sample Midterm Exam 1 Spring 2015To be permitted: one (1) 8 1/2" x 11" one sided sheet of hand-written notesNo calculators(*) = harder and/or more time consuming questionsProblem 1 ( 15 points) A particle of mass m moves in one-dimension while suffering a drag forceopposed to its motion of magnitude Cv2/3 where C is a positive constant. It starts at the origin x = 0 attime 0 with velocity vo > 0, that then diminishes due to the drag. a) What is its speed after time t? b) At what time does it stop?*c) How far does it go before stopping?Problem 2: (15 points) A particle of mass m=2kg moves in 1-d under a force distribution withpotential energy (in Joules) given by the plot (tic marks indicate Joules and meters).Describe the possible motions of the massqualitatively, and estimate the x-value of anyturning points(s), if… a) It has energy E = 4 Joules b) It has energy E = 1 Joule c) It has energy E = - 3 Joule *d) Now assume it starts at positionxo = 0 with vo = 2m/sec to the right. What isits kinetic energy there? What is its Potentialenergy there?What is its speed and position as t goesto ∞?Problem 3: (20 points) A uniform rod of total mass M and length L has negligible cross sectional area. We are interested inthe gravitational field at a point to the right of the right end. a) Find the gravitational potential Φ at a point a distance 'a' from the end of the rod. b) Show that your expression for Φ has the correct a-dependence in the limit of large a: a >> L; a < ∞Problem 4a: (10 points) Consider a uniform thin spherical shell ofradius R and mass Mshell hanging in otherwise empty space. At thecenter of the shell is a point mass MpointFind the magnitude |rg(r) | of the gravitational acceleration at pointsa distance r from the center and outside the shell r >R.Find the magnitude |rg(r) | of the gravitational acceleration at points a distance r from the center andinside the shell r < R.(*) Choose Φ(∞) = 0 and find the expression(s) for the gravitational potential Φ(r) for all points r insideand outside the shell. Make sure your Φ is continuous at R.Problem 4b (10 points) A thin circular hoop of mass Mand radius R hangs in empty space. What is thegravitational potential Φ at a point P a distance x above itscenter ? (Hint: this is very easy.)Problem 5: (30 points) Consider a spaceship of small mass µ in an elliptical orbit around the sun. Theorbit has eccentricity ε < 1 and closest approach (perihelion) a distance Rmin from the sun. Allanswers are to be given in terms of µ, and the sun's mass M, and G, ε, and the given Rmin.a) What is its furthest distance (aphelion) Rmaxfrom the sun?b) What is its speed vperihelion at closestapproach?c) What is its speed vaphelion at Rmax?d) What is the magnitude |L| of its angular momentum ?e) What is its energy E?*f) What instantaneous increase of speed Δv is required if the spaceship is to boost into an escapetrajectory from perihelion r = Rmin ?Problem 6. The gravitational potential on theaxis above a thin disk was worked out in class tobeΦ = −2πGρh[ R2+ x2− x] where ρ is thevolumetric mass density of the disk material and his its thickness.If two particles are located at differentheights, x and x + ε, where ε is small andpositive, find the tidal acceleration between them.Which one accelerates faster and by how much?Possibly Useful FormulaeCylindrical Coordinates Spherical CoordinatesddxA[B(x)] =dAdBdBdx r(φ) =α1+εcos(φ);α= l2/ GM ;ε= 1+2el2G2M2Φ(rr ) = −GV1∫ρ1(rr1)|rr
View Full Document
Unlocking...