285
REVIEW & SUM MARY
Angular Position To describe the rotation of a rigid body about
a fixed axis, called the rotation axis, we assume a reference line is
fixed in the body, perpendicular to that axis and rotating with the
body.We measure the angular position u of this line relative to a fixed
direction.When u is measured in radians,
(radian measure),
(10-1)
where s is the arc length of a circular path of radius r and angle u.
Radian measure is related to angle measure in revolutions and degrees by
1 rev ϭ 360Њ ϭ 2p rad.
(10-2)
Angular Displacement A body that rotates about a rotation
axis, changing its angular position from u 1 to u 2 , undergoes an angular displacement
⌬u ϭ u 2 Ϫ u 1 ,
( 1 0 - 4 )
where ⌬u is positive for counterclockwise rotation and negative for
clockwise rotation.
Angular Velocity and Speed If a body rotates through an
angular displacement ⌬u in a time interval ⌬t, its average angular
velocity v avg is
(10-5)
The (instantaneous) angular velocity v of the body is
(10-6)
Both v avg and v are vectors, with directions given by the right-hand
rule of Fig. 10-6. They are positive for counterclockwise rotation
and negative for clockwise rotation. The magnitude of the body’s
angular velocity is the angular speed.
Angular Acceleration If the angular velocity of a body
changes from v 1 to v 2 in a time interval ⌬t ϭ t 2 Ϫ t 1 , the average
angular acceleration a avg of the body is
(10-7)
The (instantaneous) angular acceleration a of the body is
(10-8)
Both a avg and a are vectors.
The Kinematic Equations for Constant Angular Acceleration Constant angular acceleration (a ϭ constant) is an important special case of rotational motion. The appropriate kinematic equations, given in Table 10-1, are
v ϭ v 0 ϩ at,
( 1 0 - 1 2 )
(10-13)
(10-14)
(10-15)
(10-16)
Linear and Angular Variables Related A point in a rigid
rotating body, at a perpendicular distance r from the rotation axis,
u Ϫ u 0 ϭ vt Ϫ
1
2 at
2
.
u Ϫ u 0 ϭ
1
2 (v 0 ϩ v)t,
v
2 ϭ v 0
2 ϩ 2a(u Ϫ u 0 ),
u Ϫ u 0 ϭ v 0 t ϩ
1
2 at
2
,
a ϭ
dv
dt
.
a avg ϭ
v 2 Ϫ v 1
t 2 Ϫ t 1
ϭ
⌬v
⌬t
.
v ϭ
du
dt
.
v avg ϭ
⌬u
⌬t
.
u ϭ
s
r
Review & Summary
moves in a circle with radius r. If the body rotates through an angle u,
the point moves along an arc with length s given by
s ϭ ur (radian measure),
(10-17)
where u is in radians.
The linear velocity of the point is tangent to the circle; the
point’s linear speed v is given by
v ϭ vr (radian measure),
(10-18)
where v is the angular speed (in radians per second) of the body.
The linear acceleration of the point has both tangential and
radial components. The tangential component is
a t ϭ ar (radian measure),
(10-22)
where a is the magnitude of the angular acceleration (in radians
per second-squared) of the body. The radial component of is
(radian measure).
(10-23)
If the point moves in uniform circular motion, the period T of
the motion for the point and the body is
(radian measure).
(10-19, 10-20)
Rotational Kinetic Energy and Rotational Inertia The kinetic energy K of a rigid body rotating about a fixed axis is given by
(radian measure),
(10-34)
in which I is the rotational inertia of the body, defined as
(10-33)
for a system of discrete particles and defined as
(10-35)
for a body with continuously distributed mass. The r and r i in these
expressions represent the perpendicular distance from the axis of
rotation to each mass element in the body, and the integration is carried out over the entire body so as to include every mass element.
The Parallel-Axis Theorem The parallel-axis theorem relates
the rotational inertia I of a body about any axis to that of the same
body about a parallel axis through the center of mass:
I ϭ I com ϩ Mh
2
.
( 1 0 - 3 6 )
Here h is the perpendicular distance between the two axes, and
I com is the rotational inertia of the body about the axis through the
com. We can describe h as being the distance the actual rotation
axis has been shifted from the rotation axis through the com.
Torque Torque is a turning or twisting action on a body about a rotation axis due to a force . If is exerted at a point given by the position vector relative to the axis, then the magnitude of the torque is
(10-40, 10-41, 10-39)
where F t is the component of perpendicular to and f is the angle between and . The quantity is the perpendicular distance
between the rotation axis and an extended line running through
the
vector. This line is called the line of action of , and
is
called the moment arm of . Similarly, r is the moment arm of F t .
F
:
r Ќ
F
:
F
:
r Ќ
F
:
r
:
r
:
F
:
t ϭ rF t ϭ r Ќ F ϭ rF sin f,
r
:
F
:
F
:
I ϭ ͵r
2
dm
I ϭ ͚ m i r i
2
K ϭ
1
2 Iv
2
T ϭ
2pr
v
ϭ
2p
v
a r ϭ
v
2
r
ϭ v
2
r
a
:
a
:
v
:
REVIEW & SUM MARY
Angular Position To describe the rotation of a rigid body about
a fixed axis, called the rotation axis, we assume a reference line is
fixed in the body, perpendicular to that axis and rotating with the
body.We measure the angular position u of this line relative to a fixed
direction.When u is measured in radians,
(radian measure),
(10-1)
where s is the arc length of a circular path of radius r and angle u.
Radian measure is related to angle measure in revolutions and degrees by
1 rev ϭ 360Њ ϭ 2p rad.
(10-2)
Angular Displacement A body that rotates about a rotation
axis, changing its angular position from u 1 to u 2 , undergoes an angular displacement
⌬u ϭ u 2 Ϫ u 1 ,
( 1 0 - 4 )
where ⌬u is positive for counterclockwise rotation and negative for
clockwise rotation.
Angular Velocity and Speed If a body rotates through an
angular displacement ⌬u in a time interval ⌬t, its average angular
velocity v avg is
(10-5)
The (instantaneous) angular velocity v of the body is
(10-6)
Both v avg and v are vectors, with directions given by the right-hand
rule of Fig. 10-6. They are positive for counterclockwise rotation
and negative for clockwise rotation. The magnitude of the body’s
angular velocity is the angular speed.
Angular Acceleration If the angular velocity of a body
changes from v 1 to v 2 in a time interval ⌬t ϭ t 2 Ϫ t 1 , the average
angular acceleration a avg of the body is
(10-7)
The (instantaneous) angular acceleration a of the body is
(10-8)
Both a avg and a are vectors.
The Kinematic Equations for Constant Angular Acceleration Constant angular acceleration (a ϭ constant) is an important special case of rotational motion. The appropriate kinematic equations, given in Table 10-1, are
v ϭ v 0 ϩ at,
( 1 0 - 1 2 )
(10-13)
(10-14)
(10-15)
(10-16)
Linear and Angular Variables Related A point in a rigid
rotating body, at a perpendicular distance r from the rotation axis,
u Ϫ u 0 ϭ vt Ϫ
1
2 at
2
.
u Ϫ u 0 ϭ
1
2 (v 0 ϩ v)t,
v
2 ϭ v 0
2 ϩ 2a(u Ϫ u 0 ),
u Ϫ u 0 ϭ v 0 t ϩ
1
2 at
2
,
a ϭ
dv
dt
.
a avg ϭ
v 2 Ϫ v 1
t 2 Ϫ t 1
ϭ
⌬v
⌬t
.
v ϭ
du
dt
.
v avg ϭ
⌬u
⌬t
.
u ϭ
s
r
Review & Summary
moves in a circle with radius r. If the body rotates through an angle u,
the point moves along an arc with length s given by
s ϭ ur (radian measure),
(10-17)
where u is in radians.
The linear velocity of the point is tangent to the circle; the
point’s linear speed v is given by
v ϭ vr (radian measure),
(10-18)
where v is the angular speed (in radians per second) of the body.
The linear acceleration of the point has both tangential and
radial components. The tangential component is
a t ϭ ar (radian measure),
(10-22)
where a is the magnitude of the angular acceleration (in radians
per second-squared) of the body. The radial component of is
(radian measure).
(10-23)
If the point moves in uniform circular motion, the period T of
the motion for the point and the body is
(radian measure).
(10-19, 10-20)
Rotational Kinetic Energy and Rotational Inertia The kinetic energy K of a rigid body rotating about a fixed axis is given by
(radian measure),
(10-34)
in which I is the rotational inertia of the body, defined as
(10-33)
for a system of discrete particles and defined as
(10-35)
for a body with continuously distributed mass. The r and r i in these
expressions represent the perpendicular distance from the axis of
rotation to each mass element in the body, and the integration is carried out over the entire body so as to include every mass element.
The Parallel-Axis Theorem The parallel-axis theorem relates
the rotational inertia I of a body about any axis to that of the same
body about a parallel axis through the center of mass:
I ϭ I com ϩ Mh
2
.
( 1 0 - 3 6 )
Here h is the perpendicular distance between the two axes, and
I com is the rotational inertia of the body about the axis through the
com. We can describe h as being the distance the actual rotation
axis has been shifted from the rotation axis through the com.
Torque Torque is a turning or twisting action on a body about a rotation axis due to a force . If is exerted at a point given by the position vector relative to the axis, then the magnitude of the torque is
(10-40, 10-41, 10-39)
where F t is the component of perpendicular to and f is the angle between and . The quantity is the perpendicular distance
between the rotation axis and an extended line running through
the
vector. This line is called the line of action of , and
is
called the moment arm of . Similarly, r is the moment arm of F t .
F
:
r Ќ
F
:
F
:
r Ќ
F
:
r
:
r
:
F
:
t ϭ rF t ϭ r Ќ F ϭ rF sin f,
r
:
F
:
F
:
I ϭ ͵r
2
dm
I ϭ ͚ m i r i
2
K ϭ
1
2 Iv
2
T ϭ
2pr
v
ϭ
2p
v
a r ϭ
v
2
r
ϭ v
2
r
a
:
a
:
v
: