9-18
Chapter 9
(b)
M M
drum
mass
mg h
K
K
=+
.
2
1
MM
d
r
u
m
2
mv
mg h
K
=−
and
2
drum
22
(
2
5
0
.
0
J
)
2
2(3.71 m/s )(13.2 m)
8.04 m/s
15.0 kg
K
vg
h
m
=
−
=
EVALUATE:
We did the calculations without knowing the moment of inertia
I
of the drum, or the mass and radius
of the drum.
9.71.
IDENTIFY
and
SET UP:
All points on the belt move with the same speed. Since the belt doesn&t slip, the speed of
the belt is the same as the speed of a point on the rim of the shaft and on the rim of the wheel, and these speeds are
related to the angular speed of each circular object by
.
vr
ω
=
EXECUTE:
Figure 9.71
(a)
11
1
=
1
(60.0 rev/s)(2 rad/1 rev)
377 rad/s
ωπ
==
2
1
(0.45 10 m)(377 rad/s)
1.70 m/s
−
×
=
(b)
12
vv
=
2 2
rr
=
21
2
1
( / )
(0.45 cm/2.00 cm)(377 rad/s)
84.8 rad/s
ωω
=
EVALUATE:
The wheel has a larger radius than the shaft so turns slower to have the same tangential speed for
points on the rim.
9.72.
IDENTIFY:
The speed of all points on the belt is the same, so
=
applies to the two pulleys.
SET UP:
The second pulley, with half the diameter of the first, must have twice the angular velocity, and this is
the angular velocity of the saw blade.
rad/s
30 rev/min
π
=
.
EXECUTE:
(a)
2
rad s
0.208 m
(2(3450 rev min))
75.1 m s.
30 rev min
2
v
⎛⎞
⎜⎟
⎝⎠
(b)
2
2
42
rad
rad s
0.208 m
2(3450 rev min)
5.43 10 m s ,
30 rev min
2
ar
=×
so the force holding sawdust on the blade would have to be about 5500 times as strong as gravity.
EVALUATE:
In
=
and
2
rad
=
,
must be in rad/s.
9.73.
IDENTIFY
and
SET UP:
Use Eq.(9.15) to relate
rad
a
to
and then use a constant acceleration equation to
replace .
EXECUTE:
(a)
2
rad
,
=
2
rad,1
1
,
=
2
rad,2
2
=
rad
rad,2
rad,1
2
1
()
aa
a
r
Δ= − = −
One of the constant acceleration equations can be written
2
1
2(
)
,
zz
ωα
θ
−
or
2
1
)
z
−=
−
Thus
rad
2
1
2
1
)2 (
)
,
r
α
θθ
αθθ
Δ=
−
as was to be shown.
(b)
2
rad
85.0 m/s
25.0 m/s
8.00 rad/s
2 (
)
2(0.250 m)(15.0 rad)
z
a
r
Δ−
=
−
Then
tan
(0.250 m)(8.00 rad/s )
2.00 m/s
=
EVALUATE:
2
is proportional to
z
and
0
−
so
rad
a
is also proportional to these quantities.
rad
a
increases
while
r
stays fixed,
z
increases, and
z
is positive.
IDENTIFY
and
SET UP:
Use Eq.(9.17) to relate
K
and
and then use a constant acceleration equation to replace
.
EXECUTE:
(c)
2
1
2
;
K
I
=
2
1
2
,
K
I
=
2
1
2
K
I
=
2 1
(
2
(
)
)
(
)
,
KK K I
I
I
Δ= − =
− =
−
as was to be shown.
(d)
2
2
45.0 J
20.0 J
0.208 kg m
(
)
(8.00 rad/s )(15.0 rad)
z
K
I
αθ θ
=
⋅
−
EVALUATE:
z
is positive,
increases, and
K
increases.