Chapter 10 1
Page 1
CHAPTER 10 - Rotational Motion About a Fixed Axis
1.
(
a
)
30° = (30°)(
π
rad/180°) =
π
/6 rad = 0.524 rad
;
(
b
)
57° = (57°)(
π
rad/180°) =
19
π
/60 = 0.995 rad
;
(
c
)
90° = (90°)(
π
rad/180°) =
π
/2 = 1.571 rad
;
(
d
)
360° = (360°)(
π
rad/180°) =
2
π
= 6.283 rad
;
(
e
)
420° = (420°)(
π
rad/180°) =
7
π
/3 = 7.330 rad
.
2.
The subtended angle in radians is the size of the object divided by the distance to the object:
= 2
r
Sun
/
r
;
(0.5°)(
π
rad/180°) = 2
r
Sun
/(150
×
10
6
km), which gives
r
Sun
≈
6.5
×
10
5
km
.
3.
We find the distance from
=
h
/
r
;
(7.5°)(
π
rad/180°) = (300 m)/
r
; which gives
r
=
2.3
×
10
3
m
.
4.
From the definition of angular acceleration, we have
=
∆
/
∆
t
= [(20,000 rev/min)(2
π
rad/rev)/(60 s/min) – 0]/(5.0 min)(60 s/min)
=
7.0 rad/s
2
.
5.
From the definition of angular velocity, we have
=∆
/
∆
t
, and we use the time for each hand to turn through a complete circle, 2
π
rad.
(
a
)
second
=∆
/
∆
t
= (2
π
rad)/(60 s) =
0.105 rad/s
.
(
b
)
minute
=∆
/
∆
t
= (2
π
rad)/(60 min)(60 s/min) =
1.75
×
10
–3
rad/s
.
(
c
)
hour
=∆
/
∆
t
= (2
π
rad)/(12 h)(60 min/h)(60 s/min) =
1.45
×
10
–4
rad/s
.
(
d
)
For each case, the angular velocity is constant, so the angular acceleration is
zero
.
6.
(
a
)
The Earth moves one revolution around the Sun in one year, so we have
orbit
=∆
/
∆
t
= (2
π
rad)/(1 yr)(3.16
×
10
7
s/yr) =
1.99
×
10
–7
rad/s
.
(
b
)
The Earth rotates one revolution in one day, so we have
rotation
=∆
/
∆
t
= (2
π
rad)/(1 day)(24 h/day)(3600 s/h) =
7.27
×
10
–5
rad/s
.
7.
All points will have the angular speed of the Earth:
=
∆
/
∆
t
= (2
π
rad)/(1 day)(24 h/day)(3600 s/h) = 7.27
×
10
–5
rad/s.
Their linear speed will depend on the distance from the rotation axis.
(
a
)
On the equator we have
v
=
r
Earth
= (6.38
×
10
6
m)(7.27
×
10
–5
rad/s) =
464 m/s
.
(
b
)
At a latitude of 66.5° the distance is
r
Earth
cos 66.5°, so we have
v
=
r
Earth
cos 66.5°
= (6.38
×
10
6
m)(cos 66.5°)(7.27
×
10
–5
rad/s) =
185 m/s
.
(
c
)
At a latitude of 40.0° the distance is
r
Earth
cos 40.0°, so we have
v
=
r
Earth
cos 40.0°
= (6.38
×
10
6
m)(cos 40.0°)(7.27
×
10
–5
rad/s) =
355 m/s
.
8.
The subtended angle in radians is the size of the object divided by the distance to the object.
A pencil with a
diameter of 6 mm will block out the Moon if it is held about 60 cm from the eye.
For the angle subtended we have

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