ENGI 9420 Engineering Analysis

Faculty of Engineering and Applied Science
2008 Fall

Final Examination Questions

  1.  
    1. Find the inverse of the matrix

      [ 1 0 0 ] [-1 1 1 ] [-1 0 -1 ]

      [9]

    2. Hence (or otherwise) solve the following problem:
      The frictional force F (in units of Newtons = kg m s–2) on a spherical body of radius a (m) falling with speed v (m s–1) through a viscous liquid of viscosity h (kg m–1 s–1) is of the form , where   k, x, y and z are dimensionless constants.   Find the values of   x, y and z.

      [7]

  1. A pair of simultaneous first order ODEs is defined by

    dx/dt = y ; dy/dt = x^2 - x - xy/2

    1. Find both critical points of this system of first order ODEs.

      [3]

    2. Determine the nature and stability of both critical points.

      [6]

    3. Sketch the phase portrait for the linear approximation to the non-linear system in the neighbourhoods of both critical points.
      [Note:   the general solution is not required.]

      [6]

    4. Hence sketch the phase portrait for the original non-linear system.

      [5]

  1. An initial value problem is defined by

    dy/dx = y - x , y(1) = 1

    1. Use the standard fourth order Runge-Kutta method, with a step size of h = 0.2, to find the value at x = 1.2 of the solution of this initial value problem.

      [9]

    2. Verify your solution to part (a) by solving the initial value problem analytically.

      [7]

  1.  

    1. Find the path   y = f (x)   between the points (0, 1) and (1, 1) for which the integral

      I = Integral[0 to 1] { 12xy + (y')^2 } dx

      has an extremum and determine whether the extremum is a maximum or a minimum.

      [16]

    BONUS QUESTION

    1. If the integrand   F (x, y, y' )   is a function of   y'   only, then it follows that the extremal path is the straight line joining the endpoints, unless partial d^2 F / d(y')^2 ident= 0.   Show that, if partial d^2 F / d(y')^2 ident= 0, then the integral I = Integral[x0 to x1] { F(y') } dx has the same value, no matter what path   y = f (x)   is taken between the fixed points (x0, y0) and (x0, y0)

      [+6]

  1. Sketch the graph of the odd periodic extension of the function

    f(x) = x (pi – x) , (0 <= x &<= pi)

    and find its Fourier half-range sine series expansion.

    [16]

  1. A cylinder containing a viscous fluid is rotating at a constant angular speed   wo . The fluid moves with a velocity (in the cylindrical polar coordinate system) of

    vector v = rho omega_o vector phiHat

    1. Find the divergence of the velocity vector.

      [7]

    2. Find the curl of the velocity vector.

      [6]

    3. Find d/dt curl v

      [3]

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        Created 2008 12 17 and most recently modified 2008 12 17 by Dr. G.H. George.