Ch 8 Rotational Kinematics

These are our class notes from Chapter 8 of Cutnell and Johnson's Physics

1. Rotational Kinematics
Chapters 8

2. Learning Objectives
Objectives for this chapter fall under applications
of previous objectives

3. Table Of Contents
1.Rotational Motion and Angular Displacement
2.Angular Velocity and Angular Acceleration
3.The Equations of Rotational Kinematics
4.Angular Variables and Tangential Acceleration
5.Centripetal Acceleration and Tangential
Acceleration
6.Rolling Motion
7.The Vector Nature of Angular Variables

4. Chapter 8:
Rotational Kinematics
Section 1:
Rotational Motion and
Angular Displacement

5. Describing Rotating Objects
As we began to discuss,
not all motion is linear.
In the simplest kind of
rotation, points on a rigid
object move on circular
paths around an axis of
rotation.

6. o
qqq-=D
First there was displacement (d)
Need to develop new
variables for polar
system
The angle through
which the object rotates
is called the Angular
displacement.

7. DEFINITION OF ANGULAR DISPLACEMENT
When a rigid body rotates about a fixed axis, the
angular displacement is the angle swept out by a
line passing through any point on the body and
intersecting the axis of rotation perpendicularly.
By convention, the angular displacement is
positive if it is counterclockwise and negative if it
is clockwise.
SI Unit of
Angular Displacement: radian (rad)

8. r
s
==
Radius
length Arc
radians)(in q
For a full revolution:

360rad 2 =prad 2
2
p
p
q ==
r
r

9. Example 1 Adjacent Synchronous Satellites
Synchronous satellites are put into
an orbit whose radius is 4.23×10
7
m.
If the angular separation of the two
satellites is 2.00 degrees, find the
arc length that separates them.
rad 0349.0
deg360
rad 2
deg00.2 =
÷
÷
ø
ö
ç
ç
è
æp
qrs=
r
s
==
Radius
length Arc
radians)(in q
( )( )rad 0349.0m1023.4
7
´=
miles) (920 m1048.1
6
´=s

10. 8.1.1. Over the course of a day (twenty-four hours), what is the
angular displacement of the second hand of a wrist watch in
radians?
a) 1440 rad
b) 2880 rad
c) 4520 rad
d) 9050 rad
e) 543 000 rad

11. Chapter 8:
Rotational Kinematics
Section 2:
Angular Velocity and
Angular Acceleration

12. o
qqq-=D
How do we describe the rate
at which the angular displacement
is changing?
Next, there was “velocity”

13. DEFINITION OF AVERAGE ANGULAR VELOCITY
timeElapsed
ntdisplacemeAngular
locity angular ve Average =
o
o
tt-
-
=
qq
w
SI Unit of Angular Velocity: radian per second (rad/s)
tD
D
=
q

14. Example 3 Gymnast on a High Bar
A gymnast on a high bar swings through
two revolutions in a time of 1.90 s.
Find the average angular velocity
of the gymnast.
÷
ø
ö
ç
è
æ
-=D
rev 1
rad 2
rev 00.2
p
q
s 90.1
rad 6.12-
=w
rad 6.12-=
tD
D
=
q
w
srad63.6-=

15. ww
0
lim
®D
=
t
INSTANTANEOUS ANGULAR VELOCITY
Instantaneous Angular speed
The magnitude of the instantaneous angular velocity
t
tD
D
=
®D
q
0
lim
SI Unit of Instantaneous Angular velocity: radian per second (rad/s)

16. Changing angular velocity means that an angular
acceleration is occurring.
DEFINITION OF AVERAGE ANGULAR ACCELERATION
ttt
o
o
D
D
=
-
-
=
www
a
timeElapsed
locityangular vein Change
on acceleratiangular Average =
SI Unit of Angular acceleration: radian per second per second (rad/s
2
)
Then, there was “acceleration”

17. Example 4 A Jet Revving Its Engines
As seen from the front of the
engine, the fan blades are
rotating with an angular
speed of -110 rad/s. As the
plane takes off, the angular
velocity of the blades reaches
-330 rad/s in a time of 14 s.
Find the angular acceleration, assuming it to
be constant.
( )( )
2
srad16
s 14
srad110srad330
-=
---
=a

18. 8.2.1. The planet Mercury takes only 88 Earth days to orbit the
Sun. The orbit is nearly circular, so for this exercise, assume
that it is. What is the angular velocity, in radians per second,
of Mercury in its orbit around the Sun?
a) 8.3 × 10
-7
rad/s
b) 2.0 × 10
-5
rad/s
c) 7.3 × 10
-4
rad/s
d) 7.1 × 10
-2
rad/s
e) This cannot be determined without knowing the radius of the
orbit.

19. 8.2.2. Complete the following statement: For a wheel that turns with
constant angular speed,
a) each point on its rim moves with constant acceleration.
b) the wheel turns through “equal angles in equal times.”
c) each point on the rim moves at a constant velocity.
d) the angular displacement of a point on the rim is constant.
e) all points on the wheel are moving at a constant velocity.

20. Chapter 8:
Rotational Kinematics
Section 3:
The Equations of Rotational Kinematics

21. Five kinematic variables
Making Sense of it All
displacement
initial velocity
final velocity
elapsed time
acceleration
Dx
v
o
v
t
a
Dq
w
o
w
t
a

22. t
oaww +=
( )t
owwq +=D
2
1
( )qaww D+= 2
22
o
2
2
1
tt
o
awq +=D
The equations of rotational kinematics for constant
angular acceleration:
atvv
o+=
( )tvvx
o+=D
2
1
()xavv
o
D+= 2
22
2
2
1
attvx
o
+=D

23. Reasoning Strategy
1.Make a drawing.
2.Decide which directions are to be called positive (+) and
negative (-). (Counterclockwise default +)
3.Write down the values that are given for any of the five
kinematic variables.
4.Verify that the information contains values for at least three
of the five kinematic variables. Select the appropriate
equation.
5.When the motion is divided into segments, remember that
the final angular velocity of one segment is the initial
velocity for the next.
6.Keep in mind that there may be two possible answers to a
kinematics problem.

24. Example 5 Blending with a Blender
The blades are whirling with an
angular velocity of +375 rad/s when
the “puree” button is pushed in.
When the “blend” button is pushed,
the blades accelerate and reach a
greater angular velocity after the
blades have rotated through an
angular displacement of +44.0 rad.
The angular acceleration has a
constant value of +1740 rad/s
2
.
Find the final angular velocity of the blades.

25. +375 rad/s?+1740 rad/s
2
+44.0 rad
tω
oωαDθ
( )qaww D+= 2
22
o
( )qaww D+= 2
2
o
( ) ( )( )rad0.44srad17402srad375
22
+=w
srad542+=w

26. 8.3.1. The propeller of an airplane is at rest when the pilot starts
the engine; and its angular acceleration is a constant value.
Two seconds later, the propeller is rotating at 10p rad/s.
Through how many revolutions has the propeller rotated
through during the first two seconds?
a) 300
b) 50
c) 20
d) 10
e) 5

27. 8.3.2. A ball is spinning about an axis that passes through its center
with a constant angular acceleration of p rad/s
2
. During a time
interval from t
1
to t
2
, the angular displacement of the ball is p
radians. At time t
2
, the angular velocity of the ball is 2p rad/s.
What is the ball’s angular velocity at time t
1?
a) 6.28 rad/s
b) 3.14 rad/s
c) 2.22 rad/s
d) 1.00 rad/s
e) zero rad/s

28. Chapter 8:
Rotational Kinematics
Section 4:
Angular Variables and
Tangential Acceleration

29. Relating to the “Worlds”
Sometimes it’s easier to
solve the problem with
“rotational” kinematics,
sometimes with
“translational”
What we will now look at is
relating or transforming
from one system to the
other.
How does the “linear” or
rather tangential speed
relate to the angular
speed?

30. qrs=
t
ss
v
0-
=
t
o
qq
w
-
=
wrv=
t
vv
a
0-
=
t
o
ww
a
-
=
ara=

31. Example 6 A Helicopter Blade
A helicopter blade has an angular speed of 6.50 rev/s and an
angular acceleration of 1.30 rev/s
2
.
For point 1 on the blade, find
the magnitude of (a) the
tangential speed and (b) the
tangential acceleration.
÷
ø
ö
ç
è
æ
÷
ø
ö
ç
è
æ
=
rev 1
rad 2
s
rev
50.6
p
w
arv
T=
ara
T
=
÷
ø
ö
ç
è
æ
÷
ø
ö
ç
è
æ
=
rev 1
rad 2
s
rev
30.1
2
p
a
srad 8.40=
( )( )srad8.40m 3.00= sm122=
2
srad 17.8=
( )( )
2
srad17.8m 3.00=
2
sm5.24=

32. 8.4.1. The Earth, which has an equatorial radius of 6380 km, makes
one revolution on its axis every 23.93 hours. What is the
tangential speed of Nairobi, Kenya, a city near the equator?
a) 37.0 m/s
b) 74.0 m/s
c) 148 m/s
d) 232 m/s
e) 465 m/s

33. 8.4.2. The original Ferris wheel had a radius of 38 m and completed
a full revolution (2p radians) every two minutes when operating
at its maximum speed. If the wheel were uniformly slowed from
its maximum speed to a stop in 35 seconds, what would be the
magnitude of the instantaneous tangential speed at the outer rim
of the wheel 15 seconds after it begins its deceleration?
a) 0.295 m/s
b) 1.12 m/s
c) 1.50 m/s
d) 1.77 m/s
e) 2.03 m/s

34. 8.4.3. A long, thin rod of length 4L rotates counterclockwise with
constant angular acceleration around an axis that is perpendicular to
the rod and passes through a pivot point that is a length L from one
end as shown. What is the ratio of the tangential acceleration at a
point on the end closest to the pivot point to that at a point on the end
farthest from the pivot point?
a) 4
b) 3
c) 1/2
d) 1/3
e) 1/4

35. 8.4.4. A long, thin rod of length 4L rotates counterclockwise with
constant angular acceleration around an axis that is perpendicular to
the rod and passes through a pivot point that is a length L from one
end as shown. What is the ratio of the tangential speed (at any
instant) at a point on the end closest to the pivot point to that at a
point on the end farthest from the pivot point?
a) 1/4
b) 1/3
c) 1/2
d) 3
e) 4

36. Chapter 8:
Rotational Kinematics
Section 5:
Centripetal Acceleration and
Tangential Acceleration

37. Relationship to Circular Motion
wrv=
r
v
a
c
2
=
How do we relate rotational kinematics to Uniform Circular
Motion (Ch 5)
Even if “tangential” speed is constant, there is still an
acceleration.
2
wra
c
=
( )
r
r
a
c
2
w
=

38. Tangential Acceleration?
From Ch 5, we define the axes as polar co-ordinates not
Cartesian.
(r,q) not (x,y)
Since they are perpendicular, we can treat each axis
independently
Therefore, total acceleration would be vector sum of a
c plus
a
T
22
Tcaaa +=
c
T
a
a
=ftan

39. Example 7 A Discus Thrower
Starting from rest, the thrower
accelerates the discus to a final
angular speed of +15.0 rad/s in
a time of 0.270 s before releasing it.
During the acceleration, the discus
moves in a circular arc of radius
0.810 m.
Find the total acceleration.

40. 2
wra
c=
t
ω-ω
rra
o
T
==a
22
cTaaa +=
÷
÷
ø
ö
ç
ç
è
æ
=
-
c
T
a
a
1
tanf
÷
÷
ø
ö
ç
ç
è
æ
=
2
2
182
0.45
sm
sm o
9.13=
( )( )
22
sm0.45sm182 +=
2
sm187=
( ) ÷
ø
ö
ç
è
æ
=
s 0.270
srad0.15
m 810.0
2
sm0.45=
( )( )
2
srad0.15m 810.0=
2
sm182=

41. 8.5.1. An airplane starts from rest at the end of a runway and begins
accelerating. The tires of the plane are rotating with an angular velocity
that is uniformly increasing with time. On one of the tires, Point A is
located on the part of the tire in contact with the runway surface and
point B is located halfway between Point A and the axis of rotation.
Which one of the following statements is true concerning this situation?
a) Both points have the same tangential acceleration.
b) Both points have the same centripetal acceleration.
c) Both points have the same instantaneous angular velocity.
d) The angular velocity at point A is greater than that of point B.
e) Each second, point A turns through a greater angle than point B.

42. 8.5.2. A wheel starts from rest and rotates with a constant angular
acceleration. What is the ratio of the instantaneous tangential
acceleration at point A located a distance 2r to that at point B
located at r, where the radius of the wheel is R = 2r?
a) 0.25
b) 0.50
c) 1.0
d) 2.0
e) 4.0

43. Chapter 8:
Rotational Kinematics
Section 6:
Rolling Motion

44. Rolling Motion
A rolling object is a common example of when you
would need to be able to transform from one
coordinate system to the other.
Usually a “rolling” object will not be slipping on the
surface.

45. wrv=
The tangential speed of a
point on the outer edge of
the tire is equal to the speed
of the car over the ground.
ara=
qrsd==

46. Example 8 An Accelerating Car
Starting from rest, the car accelerates
for 20.0 s with a constant linear
acceleration of 0.800 m/s
2
. The
radius of the tires is 0.330 m.
What is the angle through which
each wheel has rotated?

47. 2
2
1
tt
o
awq +=
20.0 s0 rad/s-2.42 rad/s
2
?
tω
oωαθ
2
2
srad42.2
m 0.330
sm800.0
===
r
a
a
( )( )
222
2
1
s 0.20srad42.2-=q rad 484-=

48. 8.6.1. The wheels of a bicycle have a radius of r meters. The bicycle is traveling
along a level road at a constant speed v m/s. Which one of the following
expressions may be used to determine the angular speed, in rev/min, of the
wheels?
a)
b)
c)
d)
e)
r
v
r
v
30
p
r
v

30
p
r
v
2
30
p
r
v

60
p

49. 8.6.2. Josh is painting yellow stripes on a road using a paint roller.
To roll the paint roller along the road, Josh applies a force of 15
N at an angle of 45° with respect to the road. The mass of the
roller is 2.5 kg; and its radius is 4.0 cm. Ignoring the mass of the
handle of the roller, what is the magnitude of the tangential
acceleration of the roller?
a) 4.2 m/s
2
b) 6.0 m/s
2
c) 15 m/s
2
d) 110 m/s
2
e) 150 m/s
2

50. 8.6.3. Which one of the following statements concerning a wheel
undergoing rolling motion is true?
a) The angular acceleration of the wheel must be zero m/s
2
.
b) The tangential velocity is the same for all points on the wheel.
c) The linear velocity for all points on the rim of the wheel is non-
zero.
d) The tangential velocity is the same for all points on the rim of the
wheel.
e) There is no slipping at the point where the wheel touches the
surface on which it is rolling.

51. 8.6.4. A circular hoop rolls without slipping on a flat horizontal
surface. Which one of the following is necessarily true?
a) All points on the rim of the hoop have the same speed.
b) All points on the rim of the hoop have the same velocity.
c) Every point on the rim of the wheel has a different velocity.
d) All points on the rim of the hoop have acceleration vectors that
are tangent to the hoop.
e) All points on the rim of the hoop have acceleration vectors that
point toward the center of the hoop.

52. 8.6.5. A bicycle wheel of radius 0.70 m is turning at an angular
speed of 6.3 rad/s as it rolls on a horizontal surface without
slipping. What is the linear speed of the wheel?
a) 1.4 m/s
b) 28 m/s
c) 0.11 m/s
d) 4.4 m/s
e) 9.1 m/s

53. Chapter 8:
Rotational Kinematics
Section 7:
The Vector Nature of Angular Variables

54. Here’s where is gets a little crazy!
The direction of the angular
velocity vector point along the axis
of rotation.
Right-Hand Rule: Grasp the axis
of rotation with your right hand, so
that your fingers circle wrap the
axis in the same direction as the
rotation.
Your extended thumb points along
the axis in the direction of the
angular velocity.

55. 8.7.1. A packaged roll of paper towels falls from a shelf in a grocery
store and rolls due south without slipping. What is the direction
of the paper towels’ angular velocity?
a) north
b) east
c) south
d) west
e) down

56. 8.7.2. A packaged roll of paper towels falls from a shelf in a grocery
store and rolls due south without slipping. As its linear speed
slows, what are the directions of the paper towels’ angular velocity
and angular acceleration?
a) east, east
b) west, east
c) south, north
d) east, west
e) west, west

57. 8.7.3. A top is spinning counterclockwise and moving toward the
right with a linear velocity as shown in the drawing. If the
angular speed is decreasing as time passes, what is the
direction of the angular velocity of the top?
a) upward
b) downward
c) left
d) right

58. 8.7.4. A truck and trailer have 18 wheels. If the direction of the
angular velocity vectors of the 18 wheels point 30° north of
east, in what direction is the truck traveling?
a) 30° east of south
b) 30° west of north
c) 30° north of east
d) 30° south of west
e) 30° south of east

59. 8.7.5. A girl is sitting on the edge of a merry-go-round at a
playground as shown. Looking down from above, the merry-
go-round is rotating clockwise. What is the direction of the
girl’s angular velocity?
a) upward
b) downward
c) left
d) right
e) There is no direction since it is the merry go round that has the
angular velocity.