Solution Derivations for Capa #10
1) The flywheel of a steam engine runs with a constant angular speed of 172
rev/min
.
When steam is shut off, the friction of the bearings and the air brings the wheel to rest in 1
.
0
hours. What is the magnitude of the constant angular acceleration of the wheel? (Answer
in rev/min
2
)
ω
0
= Given
t
= Given in hours, convert to minutes.
Since
ω
is given in
rev
min
, we can directly substitute into the angular kinematics
equations. For this problem,
ω
=
ω
0
+
αt
comes in handy. Solving for
α
,
α
=
ω

ω
0
t
In this case,
ω
(the final rotational speed) is zero since the engine flywheel stops.
So,
α
=

ω
0
t
.
Units are
rev
min
2
and CAPA is looking for the magnitude of the answer.
2) How many rotations does the wheel make before coming to rest? (No units required)
For this problem, remember from translational kinematics that
x
= ¯
vt
. Similarly,
in rotational kinematics,
θ
= ¯
ωt
. Average angular speed is given by
1
2
(
ω
+
ω
0
).
Thus, the equation becomes
θ
=
1
2
(
ω
+
ω
0
)
t
Where
ω
is zero since the flywheel comes to rest and
t
is in minutes. Simplifying,
θ
=
1
2
ω
0
t
3) What is the magnitude of the tangential component of the linear acceleration of a
particle that is located at a distance of 37
cm
from the axis of rotation when the flywheel is
turning at 86
rev/min
?
This problem may be a little confusing because it does not say that it is still
related to problem 1. In this problem you are asked to find the tangential com
ponent of the acceleration. This is given by
a
t
=
rα
But
α
was found in the first problem in units
rev
min
2
.
The radius is given in
this problem in cm. Note that the angular speed given has no meaning in this
1
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problem. First convert the radius to meters. Then convert
α
to radians. Finally
convert the minutes to seconds. (This last step may be omitted because CAPA
knows all units. However, I have not tried this). This can all be done with the
following step
a
t
=
r
*
1
m
100
cm
*
α
*
2
π rad
rev
*
1
min
60
sec
2
or simply
a
t
=
r
100
2
πα
3600
=
rπα
180 000
Remember the sign on
α
; units, of course, will be in
m/s
2
. All of the conversions
are built into the formula. Thus, you would enter
r
in
cm
and
α
in
rev/min
2
(your answer from (1) )
4) What is the magnitude of the net linear acceleration of the particle in the above
question?
In this problem, you are asked to find the net linear acceleration which means you
need to find the other component of the acceleration. Acceleration is a vector and
is composed of both tangential and radial components. The radial component of
acceleration is
a
r
=
v
2
r
But
v
=
ωr
, so
a
r
=
w
2
r
2
r
=
ω
2
r
In question 3,
ω
is given in revolutions per minute, so you must convert to radians
per second. This can be done by the following:
a
r
=
ω
2
*
2
π rad
rev
2
*
r
*
1
m
100
cm
*
1
min
60
sec
2
or
a
r
=
ω
2
*
4
π
2
*
r
100
*
1
3600
=
ω
2
π
2
r
90 000
However, CAPA is asking for the net acceleration. To get this just add
a
r
and
a
t
.
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 Spring '08
 SPIKE,BENJ
 Physics, Energy, Friction, Kinetic Energy, Moment Of Inertia

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