189
7
CHAPTER
Rotational Motion and the
Law of Gravity
7.1
Angular Speed and
Angular Acceleration
7.2
Rotational Motion under
Constant Angular
Acceleration
7.3
Relations between
Angular and Linear
Quantities
7.4
Centripetal Acceleration
7.5
Newtonian Gravitation
7.6
Kepler’s Laws
Courtesy NASA
Rotational motion is an important part of everyday life. The rotation of the Earth creates the
cycle of day and night, the rotation of wheels enables easy vehicular motion, and modern
technology depends on circular motion in a variety of contexts, from the tiny gears in a Swiss
watch to the operation of lathes and other machinery. The concepts of
angular speed, angu
lar acceleration
, and
centripetal acceleration
are central to understanding the motions of a di
verse range of phenomena, from a car moving around a circular race track to clusters of
galaxies orbiting a common center.
Rotational motion, when combined with Newton’s law of universal gravitation and
his laws of motion, can also explain certain facts about space travel and satellite motion,
such as where to place a satellite so it will remain Fxed in position over the same spot on
the Earth. The generalization of gravitational potential energy and energy conservation
offers an easy route to such results as planetary escape speed. ±inally, we present Kepler’s
three laws of planetary motion, which formed the foundation of Newton’s approach to
gravity.
7.1
ANGULAR SPEED AND ANGULAR
ACCELERATION
In the study of linear motion, the important concepts are
displacement
D
x
,
velocity v
,
and
acceleration a
. Each of these concepts has its analog in rotational motion:
angu
lar displacement
D
u
,
angular velocity
v
, and
angular acceleration
a
.
The
radian
, a unit of angular measure, is essential to the understanding of these
concepts. Recall that the distance
s
around a circle is given by
s
±
2
p
r
, where
r
is
Astronauts F. Story Musgrave and
Jeffrey A. Hoffman, along with the
Hubble Space Telescope and the
Space Shuttle
Endeavor,
are all
“falling” around Earth.
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Chapter 7
Rotational Motion and the Law of Gravity
the radius of the circle. Dividing both sides by
r
results in
s/r
±
2
p
. This quantity is
dimensionless, because both
s
and
r
have dimensions of length, but the value 2
corresponds to a displacement around a circle. A half circle would give an
answer of
, a quarter circle an answer of
/2. The numbers 2
,
, and
/2 corre
spond to angles of 360
8
, 180
8
, and 90
8
, respectively, so a new unit of angular
measure, the
radian
, can be deFned as
the arc length
s
along a circle divided by the
radius
r
:
[7.1]
±igure 7.1 illustrates the size of 1 radian, which is approximately 53
8
. ±or conver
sions, we use the fact that 360
8±
2
radians (or 180
radians). ±or example,
45
8
(2
rad/360
8
)
±
(
/4) rad.
Generally, angular quantities in physics must be expressed in radians. Be
sure to set your calculator to radian mode; neglecting to do this is a common
error.
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 Spring '08
 Turner
 Acceleration, Force, Gravity, Rotation, Velocity

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