130
THE DIRAC &FUNCTION
12.
An
infinitely long, one-dimensional wire with
a
constant mass per unit length
A,
is bent to follow the curve y
=
sinx
in
the
z
=
0
plane.
Determine the mass density
p,(x,
y,
z)
that describes
this
mass distribution.
13.
A wire of mass per unit length
A,
is bent to follow the shape of a closed ellipse
that lies in the xy-plane and is given by the expression
x2
+
2y2
=
4.
Express
p,(x,y,z),
the mass per unit volume of
this
object, using Dirac
6-
functions.
Show
that your expression has the proper dimensions. There
is
more
than one way to express the answer to
this
problem. Identify the most compact
form.
14.
An infinite, one-dimensional bar of mass per
unit
length
A,
lies along the line
y
=
m,x
in the
z
=
0
plane.
Y
(a)
Determine the mass per unit volume
p(x,
y,
z)
of
this
bar.
(b)
Now consider the situation where
the
bar is rotating about the z-axis at
a
constant angular velocity
o,
so
that the angle the bar makes with respect to
the x-axis is given by
8
=
mot,
as shown below. Find
an
expression for the
time-dependent mass density
p(x,
y,
z,
t).
