Physics 325 Quiz #2  Solutions
Wednesday 25 October 2006
There are 30 points for each correct question for a total of 60 points.
1.
A particle of mass
m
moves in two dimensions under the following potential energy
function:
U
(
r
) =
k
(
x
2
+ 4
y
2
)
/
2
a)
Find the force
F
(
r
).
The force is given by
F
(
r
) =

V
, so we have
F
(
r
) =

V
=

∂
∂x
V
(
r
)
ˆ
i
 
∂
∂y
V
(
r
)
ˆ
j
=

kx
ˆ
i

4
ky
ˆ
j
b)
Calculate the work done in moving the mass from the point (0
,
0) to the point (
a, b
)
using the following two paths:
b.i)
from
x
= 0 to
x
=
a
along
y
= 0, followed by
y
= 0 to
y
=
b
along
x
=
a
.
Call point (0
,
0) =
A
and call point (
a, b
) =
B
, then we have
W
AB
=
A
F
(
r
)
·
dr
with
dr
=
dx
ˆ
i
+
dy
ˆ
j
For the first path, we have
W
AB
=
a
0

kxdx
+
b
0

4
kydy
=

k
x
2
2

a
0

2
ky
2

b
0
=

ka
2
2

2
kb
2
b.ii)
from (0
,
0) to (
a, b
) along a direct line between these two points.
For this path, we need to introduce parametric equations.
Let
x
=
at
and
y
=
bt
, so we have
dx
=
adt
and
dy
=
bdt
and we are integrating from
t
= 0 to
t
= 1:
W
AB
=
(
a,b
)
(0
,
0)

kxdx

4
kydy
=
t
=1
t
=0

ka
2
tdt

4
kb
2
tdt
=

ka
2
2

2
kb
2
So the work done in moving a mass along these two paths is the same. This is insufficient to prove
that the force is conservative, though.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
c)
Determine mathematically if the force is conservative.
This is the end of the preview.
Sign up
to
access the rest of the document.