Physics 325, Fall 2010
Homework Solutions #4
1. (15 points)
1) a)
vector
F
(
x, y, z
) =
α
(
9
x
2
y
2
z
−
2
xyz
3
)
ˆ
x
+
α
(
6
x
3
yz
−
x
2
z
3
)
ˆ
y
+
α
(
3
x
3
y
2
−
3
x
2
z
2
y
)
ˆ
z
vector
∇ ×
vector
F
=
α
(
6
x
3
y
−
3
x
2
z
2
−
6
x
3
y
+ 3
x
2
z
2
)
ˆ
x
+
α
(
9
x
2
y
2
−
6
xyz
2
−
9
x
2
y
2
+ 6
xyz
2
)
ˆ
y
+
α
(
18
x
2
yz
−
2
xz
3
−
18
x
2
yz
+ 2
xz
3
)
ˆ
z
=
vector
0
b) Using the path described in class,
U
(
x, y, z
) =
I
x
+
I
y
+
I
z
where
I
x
=
−
integraldisplay
x
0
vector
F
(
x
′
,
0
,
0)
·
ˆ
xdx
′
= 0
It’s zero because
z
= 0 along this path.
I
y
=
−
integraldisplay
y
0
vector
F
(
x, y
′
,
0)
·
ˆ
ydy
′
= 0
It’s zero because
z
= 0 along this path.
I
z
=
−
integraldisplay
z
0
vector
F
(
x, y, z
′
)
·
ˆ
zdz
′
=
α
(
yx
2
z
3
−
3
x
3
y
2
z
)
So
U
(
x, y, z
) is just
I
z
plus a constant. But this constant needs to be zero in order for
U
(0
,
0
,
0) = 0.
U
(
x, y, z
) =
α
(
yx
2
z
3
−
3
x
3
y
2
z
)
c) The bead feels the force
F
as well as any normal forces from the wire.
The normal
forces from the wire do no work, and so the change in potential energy of the bead is minus
the change in kinetic energy.
U
(0
,
0
,
0) = 0 and
U
(
d, d, d
) =
−
2
αd
6
. So
T
(0
,
0
,
0) =
1
2
mv
2
0
and
T
(
d, d, d
) =
T
(0
,
0
,
0)
−
U
(
d, d, d
) +
U
(0
,
0
,
0) =
1
2
mv
2
0
+ 2
αd
6
. Solving for
v
(
d, d, d
) =
radicalbig
2
T
(
d, d, d
)
/m
=
radicalbig
v
2
0
+ 4
αd
6
/m
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2. (15 points)
A block of mass
m
slides down a tilted surface. The angle subtended by the surface and
the horizontal (the positive
x
axis) is
θ
. The force of gravity is directed downwards along
the negative
y
axis. The force of friction with the surface is assumed negligible. The force
of air resistance acting on the sliding block is
not
negligible and it is equal to
vector
F
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Physics, mechanics, Energy, Force, Kinetic Energy, Potential Energy, Work, vy, mg sin

Click to edit the document details