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Chemistry 120A Problem Set 1
(due January 27, 2010)
3. In class we showed that for one dimension,
dE/dt
= 0 implies
˙
p
=

dV/dx
i.e.,
f
=
ma
. Now consider many dimensions. Let
x
i
denote the Cartesian coordi
nate for dimension
i
,
i
= 1
,
2
,...,N
. The total potential energy is
V
(
x
1
,x
2
,...,x
N
),
and the total kinetic energy is
N
X
i
=1
p
2
i
/
2
m
i
,
where
m
i
is the mass associated with dimension
i
.
(a) Show that
˙
p
i
=

∂V/∂x
i
(b) For one particle in the three dimensions x, y and z, show that (a) implies
˙
p
=
∇
V
4. Consider two particles with masses
m
1
and
m
2
, and positions
r
1
and
r
2
. It is often
convenient to describe this system with the relative coordinates
r
=
r
2

r
1
and
the center of mass position
R
= (
m
1
r
1
+
m
2
r
2
)
/M
, where
M
=
m
1
+
m
2
is the net
mass. For a classical system, you can also consider the associated velocities
v
= ˙
r
and
v
cm
=
˙
R
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 Spring '10
 DAVIDCHANDLER

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