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Homework 1: # 1, 2, 6, 8, 14, 20
Michael Good
August 22, 2004
1. Show that for a single particle with constant mass the equation of motion
implies the follwing diﬀerential equation for the kinetic energy:
dT
dt
=
F
·
v
while if the mass varies with time the corresponding equation is
d
(
mT
)
dt
=
F
·
p
.
Answer:
dT
dt
=
d
(
1
2
mv
2
)
dt
=
m
v
·
˙
v
=
m
a
·
v
=
F
·
v
with time variable mass,
d
(
mT
)
dt
=
d
dt
(
p
2
2
) =
p
·
˙
p
=
F
·
p
.
2. Prove that the magnitude R of the position vector for the center of mass
from an arbitrary origin is given by the equation:
M
2
R
2
=
M
X
i
m
i
r
2
i

1
2
X
i,j
m
i
m
j
r
2
ij
.
Answer:
M
R
=
X
m
i
r
i
M
2
R
2
=
X
i,j
m
i
m
j
r
i
·
r
j
Solving for
r
i
·
r
j
realize that
r
ij
=
r
i

r
j
. Square
r
i

r
j
and you get
1
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2
ij
=
r
2
i

2
r
i
·
r
j
+
r
2
j
Plug in for
r
i
·
r
j
r
i
·
r
j
=
1
2
(
r
2
i
+
r
2
j

r
2
ij
)
M
2
R
2
=
1
2
X
i,j
m
i
m
j
r
2
i
+
1
2
X
i,j
m
i
m
j
r
2
j

1
2
X
i,j
m
i
m
j
r
2
ij
M
2
R
2
=
1
2
M
X
i
m
i
r
2
i
+
1
2
M
X
j
m
j
r
2
j

1
2
X
i,j
m
i
m
j
r
2
ij
M
2
R
2
=
M
X
i
m
i
r
2
i

1
2
X
i,j
m
i
m
j
r
2
ij
6. A particle moves in the xy plane under the constraint that its velocity vector
is always directed toward a point on the x axis whose abscissa is some given
function of time
f
(
t
). Show that for
f
(
t
) diﬀerentiable, but otherwise arbitrary,
the constraint is nonholonomic.
Answer:
The abscissa is the xaxis distance from the origin to the point on the xaxis
that the velocity vector is aimed at. It has the distance
f
(
t
).
I claim that the ratio of the velocity vector components must be equal to
the ratio of the vector components of the vector that connects the particle to
the point on the xaxis. The directions are the same. The velocity vector
components are:
v
y
=
dy
dt
v
x
=
dx
dt
The vector components of the vector that connects the particle to the point
on the xaxis are:
V
y
=
y
(
t
)
V
x
=
x
(
t
)

f
(
t
)
For these to be the same, then
v
y
v
x
=
V
y
V
x
2
dy
dx
=
y
(
t
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This note was uploaded on 03/17/2012 for the course PHYS 202 taught by Professor Atkin during the Spring '12 term at Amity University.
 Spring '12
 atkin
 Energy, Kinetic Energy, Mass, Work

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