Page 215 - Intro to Tensor Calculus

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GMm
~
~ denote the inverse square law force of attraction between the earth and sun, with
I 25. Let F = − r
r 3
G a universal constant, M the mass of the sun, m the mass of the earth and ~ r a unit vector from origin
r
at the center of the sun pointing toward the earth. (a) Write down Newton’s second law, in both vector
d
~
r
and tensor form, which describes the motion of the earth about the sun. (b) Show that (~ × ~v)= 0and
dt
~
r
r
consequently ~ × ~v = ~ × d~ r = h =aconstant.
dt
I 26. Construct a set of axes fixed and attached to an airplane. Let the x axis be a longitudinal axis running
from the rear to the front of the plane along its center line. Let the y axis run between the wing tips and
let the z axis form a right-handed system of coordinates. The y axis is called a lateral axis and the z axis is
called a normal axis. Define pitch as any angular motion about the lateral axis. Define roll as any angular
motion about the longitudinal axis. Define yaw as any angular motion about the normal axis. Consider two
sets of axes. One set is the x, y, z axes attached to and moving with the aircraft. The other set of axes is
denoted X, Y, Z and is fixed in space ( an inertial set of axes). Describe the pitch, roll and yaw of an aircraft
with respect to the inertial set of axes. Show the transformation is orthogonal. Hint: Consider pitch with

respect to the fixed axes, then consider roll with respect to the pitch axes and finally consider yaw with
respect to the roll axes. This produces three separate transformation matrices which can then be combined
to describe the motions of pitch, roll and yaw of an aircraft.

3
2
1
i
i
I 27. In Cartesian coordinates let F i = F i (x ,x ,x ) denote a force field and let x = x (t) denote a curve
!
i 2 i
d 1 dx 1 2 3 dx
C. (a) Show Newton’s second law implies that along the curve C m = F i (x ,x ,x )
dt 2 dt dt
(no summation on i) and hence
" !#
1 2 2 2 3 2 1 2 3
d 1 dx dx dx d 1 2 dx dx dx
m + + = mv = F 1 + F 2 + F 3
dt 2 dt dt dt dt 2 dt dt dt
i
i
(b) Consider two points on the curve C,say point A, x (t A ) and point B, x (t B ) and show that the work
done in moving from A to B in the force field F i is
t B Z B
1 2 1 2 3
mv = F i dx + F 2 dx + F 3 dx
2 A
t A
where the right hand side is a line integral along the path C from A to B. (c) Show that if the force field is
2
1
3
derivable from a potential function U(x ,x ,x ) by taking the gradient, then the work done is independent
of the path C and depends only upon the end points A and B.
I 28. Find the Lagrangian equations of motion of a spherical pendulum which consists of a bob of mass m
suspended at the end of a wire of length `, which is free to swing in any direction subject to the constraint
that the wire length is constant. Neglect the weight of the wire and show that for the wire attached to the
origin of a right handed x, y, z coordinate system, with the z axis downward, φ the angle between the wire
and the z axis and θ the angle of rotation of the bob from the y axis, that there results the equations of
2 2
d 2 dθ d φ dθ g
motion sin φ =0 and − sin φ cos φ + sin φ =0
dt dt dt 2 dt `