ENGI 5432 Advanced Calculus

Faculty of Engineering and Applied Science
2009 Winter

Problem Set 5 - Questions

[Sections 1.7, 2.1-2.3]

Find the work done when an object is moved along the curve of intersection C of the circular paraboloid z = x ² + y ² and the plane 2x + y = 2 from (1, 0, 1) to (0, 2, 4) by a force .

Find the work done in travelling in once anti-clockwise around the unit circle C, centered on the origin, in the presence of the force , using both of the following methods:
1. by direct evaluation of the line integral $Integral_C {F dot dr}$ and
2. by using Green’s theorem and evaluating .
3. Is the vector field F conservative?

In class we saw that the line integral $Integral_C {F dot dr}$ has the value 2p, where and the path C is one anti-clockwise circuit of the unit circle centered on the origin.
1. Show that the value of the line integral remains 2p when the vector field is replaced by
  $F = < -y / sqrt{x^2+y^2}^n, x / sqrt{x^2+y^2}^n >$
2. Show that the value of (where D is the circular area enclosed by C) is not 2p unless n < 2.
  
  Hint: transform the area integral into plane polar coordinates, with
  x = r cos q, y = r sin q and dA = dx dy = r dr dq.
3. Parts (a) and (b) clearly show that Green’s theorem does not apply when n > 2.
  Why should Green’s theorem not be used when n > 0 ?

Find a potential function for the vector field
Use this potential function to evaluate the line integral $Integral_C {F dot dr}$ for any piecewise smooth simple curve C from the point (1, 1, -1) to the point (2, 0, 5).

In the evaluation of an area integral or a volume integral , the Cartesian differentials are related to the differentials of a different parameterization (u, v, w) of the surface or volume by the Jacobian determinant:

and ,
where the Jacobian is
.

Evaluate the Jacobian in order to determine the conversion formula from Cartesian differentials dx dy (or dx dy dz) to
1. Plane polar area differentials dr dq
2. Cylindrical polar volume differentials dr df dz
3. Spherical polar volume differentials dr dq df
4. Parabolic area differentials du dv, where x = u v, y = v² - u².
5. In part (d), sketch on the same x-y plane any three members of each of the two families of coordinate curves u = constant and v = constant.

A thin wire of line density r = (ax + b) e ^z is laid out along a circle, centre the origin, radius r, in the x-y plane, (where a and b are any real constants and r is any positive real constant).
1. Show that a requirement that all parts of the wire have positive mass leads to the constraint b > | a | r.
2. Find the mass of the wire (as a function of a, b and r ).
3. Show that the centre of mass is at the point .
4. With the requirement of part (a) in place, what are the maximum and minimum possible distances of the centre of mass from the origin?
5. The moment of inertia I of a body about an axis of rotation L is defined by
  DI = r ² Dm
  (where r is the distance of the element of mass Dm from the axis L).
  [The moment of inertia is thus a second moment.]
  Find the moment of inertia of the thin wire about the z-axis.
6. The kinetic energy E of a rigid body rotating with angular velocity w about the axis L is given by E = ½ Iw². Find the kinetic energy of the thin wire when it rotates about the z-axis with an angular velocity w.
7. The angular momentum (or moment of momentum) of a rigid body rotating with angular velocity w about the axis L is Iw. [In the absence of any friction or externally applied torque, the angular momentum is conserved.] Find the angular momentum of the thin wire when it rotates about the z-axis with an angular velocity w.

Find the mass and centre of mass of a thin wire that is stretched along a straight line between the origin and the point (6, 6, 6), given that the line density at (x, y, z) is (x + y + z ) / 100 kg m^-1.

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Created 2008 01 01 and most recently modified 2008 12 27 by Dr. G.H. George