Physics 409/Classical Mechanics
Test #1
16 September 2010
Name
ANSWERS
Answer all parts of each of the two questions.
50
points total. The points value of each part is
indicated.
Begin the answer to each question on a new sheet of paper.
Clearly define all
coordinates and variables. Be sure to include sufficient detail in each answer so that your logic is
clear to me.
Useful reminders:
One form of Lagrange’s equations of motion is
−
=
=
•
∑
i
r
F
±
j
i
i
j
j
j
d
T
T
Q
dt
q
q
q
∂
∂
∂
∂
∂
∂
Another form is
where
L = T  V
0
−
=
±
j
j
d
L
L
dt
q
q
∂
∂
∂
∂
(1)(25 pts.
) A particle of mass
m
slides without friction on a very long, rigid, massless wire that
is free to rotate about the origin in the
xy
plane.
No
forces act on the particle.
(a)(
5 pts.
) Write the kinetic energy
T
of this system using plane, polar coordinates as generalized
coordinates. (
x = r
cos
θ
,
y = r
sin
θ
)
2
2
2
(
/ 2)(
)
T
m
r
r
θ
=
+
±
±
(b)(
8 pts.
) Find the equations of
motion (EOM) for this mechanical system.
2
0,
r
r
θ
−
=
±
±±
2
0
r
r
θ
θ
+
=
±±
±
±
(c)(
5 pts.
) Find an explicit, but general expression for d
T
/d
t
. Use the EOM to simplify this
expression. Is the kinetic energy
T
of the particle conserved?
2
2
/
(
)
dT
dt
m rr
rr
r
θ
θθ
=
+
+
±
±±±
±±±
±
2
2
(
[ 2
])
0
m rr
rr
rr
θ
θ
θ
θ
=
+
+
−
=
±
±
±
±
±
±
±
T
is conserved.
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 Fall '10
 Wilemski
 mechanics, Cartesian Coordinate System, Sin, Coordinate system, Polar coordinate system, Coordinate systems

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