Page 91 - Introduction to Continuum Mechanics

76 Problems

Let a coordinate frame e 1 ,C2,e 3 be attached to a rigid body which is spinning with an angular
velocity ft». Then, the angular momentum vector H c, in respect to the mass center, is given by

and

(d) Let to = a)ft and demonstrate that

and that

2C1. Prove the identities (2C1.2a-e) of Section 2C1.
2
2C2. Consider the scalar field defined by <f) = x +3xy+2z.
(a) Find a unit normal to the surface of constant 0 at the origin (0,0,0).

(b) What is the maximum value of the directional derivative of 0 at the origin
(c) Evaluate ctyldr at the origin if dr = ds(ei+e^).
2
2
2
2
2
2
2C3. Consider the ellipsoid defined by the equationx /a +y /b +z /c -l.
Find the unit normal vector at a given position (xyz).
2C4. Consider a temperature field given by 6 = 3xy.
(a) Find the heat flux at the point,4(l,l,l) if q = ~kV6.
(b) Find the heat flux at the same point as part (a) if q = -KV0, where

2C5. Consider an electrostatic potential given by 0 = a[jecos#+,ysin#j, where a and 0 are
constants.
(a) Find the electric field E if E = - V0.

(b) Find the electric displacement D if D = eE, where the matrix of e is

(c) Find the angle 0 for which the magnitude of D is a maximum.