DATE:
October 9, 2011
DUE:
October 14, 2011
ENG102, Solution to homework set #3.
1.
Look at Problem 3/101 in the book.
a. Solve the problem in the book and give the angle traveled by the mass
m
.
The angle
θ
traveled is related to the distance
s
by
s
=
rθ
.
We start by calculating the force
F
acting on
m
. This consists of two compo
nents, the horizontal force necessary for keeping the mass in a circular orbit and
a vertical necessary for keeping the mass from falling:
F
=
q
(
mg
)
2
+
m
(
v
2
/r
)
2
(1)
The acceleration of the particle along its circular path is therefore
a
t
=

μ
k
F/m
(2)
From our standard relationship
vdv
=
ads
, we then have the relationship be
tween distance
s
1
and velocity
v
1
, given an initial velocity
v
0
:
vdv
=
a
t
ds
(3)
vdv
=

μ
k
q
(
g
)
2
+ (
v
2
/r
)
2
ds
(4)

v
q
g
2
+ (
v
2
/r
)
2
dv
=
μ
k
ds
(5)
Z
v
1
v
0

v
q
g
2
+ (
v
2
/r
)
2
dv
=
μ
k
Z
s
1
0
ds
(6)
Z
v
2
0
/rg
v
2
1
/rg
1
q
1 + (
v
2
/rg
)
2
d
(
v
2
rg
)
=
2
μ
k
r
s
1
(7)
The integral on the left hand side is known to be the inverse hyperbolic sine,
sinh

1
x
= ln(
x
+
√
1 +
x
2
)
, so:
2
μ
k
r
s
1
=
ln
v
0
√
rg
2
+
r
v
0
√
rg
4
+ 1
v
1
√
rg
2
+
r
v
1
√
rg
4
+ 1
(8)
The answer to the problem is found for
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 Fall '08
 Eke
 Force, Friction, Mass, vdv

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