PHY 3221: Mechanics I
Fall Term 2010
Exam 1, September 29 2010
•
This is a closed book exam lasting 50 minutes.
•
There are three problems worth a total of 20 pts. Begin each problem on a fresh sheet of
paper. Use only one side of the paper. Avoid microscopic handwriting.
•
Put your name, the problem number and the page number in the upper righthand corner
of each sheet.
•
To receive partial credit you must explain what you are doing. Carefully labelled figures
are important! Randomly scrawled equations are not helpful.
•
Draw a box around important results (or at least results which you think might be im
portant).
•
Good luck!
1
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Problem 1.
Conservative forces.
[4 pts] Consider the following force:
F
x
=
y
,
F
y
=
x
,
F
z
=
z
. Is this a conservative force? If so, find the potential energy
U
(
x, y, z
) associated with
it.
Solution
To check if the force is conservative, calculate
vector
∇ ×
vector
F
:
(
vector
∇ ×
vector
F
)
x
=
∂F
y
∂z
−
∂F
z
∂y
= 0
−
0 = 0
.
(
vector
∇ ×
vector
F
)
y
=
∂F
z
∂x
−
∂F
x
∂z
= 0
−
0 = 0
.
(
vector
∇ ×
vector
F
)
z
=
∂F
x
∂y
−
∂F
y
∂x
= 1
−
1 = 0
.
Therefore
vector
∇ ×
vector
F
= 0 and the force is conservative. It can be represented as
vector
F
=
−∇
U
(
x, y, z
)
F
x
=
y
=
−
∂U
∂x
=
⇒
U
(
x, y, z
) =
−
yx
+
C
1
(
y, z
)
F
y
=
x
=
−
∂U
∂y
=
x
−
∂C
1
∂y
=
⇒
∂C
1
∂y
= 0 =
⇒
C
1
(
y, z
) =
C
2
(
z
) =
⇒
U
(
x, y, z
) =
−
yx
+
C
2
(
z
)
F
z
=
z
=
−
∂U
∂z
=
−
∂C
2
∂z
=
⇒
∂C
2
∂z
=
−
z
=
⇒
C
2
(
z
) =
−
1
2
z
2
+
const
The end result is
U
(
x, y, z
) =
−
xy
−
1
2
z
2
+
const
Check: the negative of the gradient of this function should reproduce the given force.
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 Spring '08
 CHAN
 mechanics, Force, Potential Energy, pts, mg cos, Vtop

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