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102
Chapter 4
•
Radian Measure
§4.4
Example 4.17
A gear with an outer radius of
r
1
=
5 cm moves in the clockwise direction, causing an interlocking
gear with an outer radius of
r
2
=
4 cm to move in the counterclockwise direction at an angular speed
of
ω
2
=
25 rpm. What is the angular speed
ω
1
of the larger gear?
r
1
=
5 cm
r
2
=
4 cm
ω
2
=
25 rpm
Figure 4.4.2
Solution:
Imagine a particle on the outer radius of each gear.
After the gears have rotated for a period of time
t
>
0, the cir
cular displacement of each particle will be the same. In other
words,
s
1
=
s
2
, where
s
1
and
s
2
are the distances traveled by
the particles on the gears with radii
r
1
and
r
2
, respectively.
But
s
1
=
ν
1
t
and
s
2
=
ν
2
t
, where
ν
1
and
ν
2
are the linear
speeds of the gears with radii
r
1
and
r
2
, respectively. Thus,
ν
1
t
=
ν
2
t
⇒
ν
1
=
ν
2
,
so by formula (4.10) we get the fundamental relation between
the two gears:
ω
1
r
1
=
ω
2
r
2
(4.11)
Note that this holds for any two gears. So in our case, we have
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.
 Fall '11
 Dr.Cheun
 Calculus

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