CHAPTER 8
Rotational Kinematics
Intro to Rotational Motion

Go to this page on your laptop or
computer:
◦ https://phet.colorado.edu/en/simulation/ladybu
g-motion-2d
◦ Click on Run Now (or Download if it is your
computer and you want to have a copy of the
app.)

Click on the circular or ellipse motions
◦ Make note of the direction of the velocity and
acceleration vectors

Consider the motion of a
rigid body about a fixed
axis
Angular Position

On your Ladybug app, click on
the handle on the turntable
with your mouse and move it
to various positions on the
circle, and observe the effect
on angle.
◦ If one segment of the turntable
rotates 50 degrees, how far would
the other segments rotate?

Move the bug to various
positions, both along a radius
and along a circle.
◦ Does the angle change when you
move along a radius?
q
Angular Position – what we need to
know




All the points in the object
maintain the same relative
position.
When the object rotates, every
point rotates through the same
angle.
The angle θ gives the angular
position of every point in the object.
When the object rotates, each
point undergoes the same angular
displacement Δθ.
q
Rotational Quantities
When an object spins, it is said to undergo
rotational motion
 The axis of rotation is the line about which the
rotation occurs.
 It is difficult to describe the motion of a point
moving in a circle using only linear quantities
because the direction of motion in a circular
path is constantly changing. For this reason,
circular motion is described in terms of the
angle through which the point on an object
moves.

Rotational Quantities

Angles can be measured in radians
◦ Radians – an angle whose arc length is equal to its
radius
◦ In general, any angle θ measured in radians is
defined by the following:
◦𝜃=
𝑠
𝑟
 s = arc length
 r = length of radius
 Θ = angle of rotation
Rotational Quantities
The angle of 360o is one revolution. (1 rev = 360o)
 One revolution is equal to the circumference of the circle of
rotation.
 Circumference is 2pr

𝑠
2𝜋𝑟

Therefore: 𝜃 = =
= 2π rad
𝑟
𝑟
so,
1 rev = 360o = 2p rad

Converting angular displacement to radians:
◦ Δ𝜃 (𝑟𝑎𝑑) =

𝜋
180
x Δθ(𝑑𝑒𝑔)
Converting radians to degrees:
◦ Δ𝜃 (deg) =
180
𝜋
x Δθ(𝑟𝑎𝑑)
8.1 Rotational Motion and Angular Displacement
Arc length s
q (in radians) 

Radius
r
For a full revolution:
q
2p r
 2p rad
r
2p rad  360
Angular displacement (Dq)


Angular displacement describes how far an object has
rotated
It is defined as:
◦ The angle through which a point, line, or body is rotated in a
specified direction and about a specified axis (in radians)
◦ Angular displacement =
◦ ∆𝜃 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
𝑟𝑎𝑑𝑖𝑢𝑠
∆𝑠
𝑟
◦ Units: radians


Counterclockwise (CCW) rotation is considered (+)
Clockwise (CW) rotation is considered (-)
Δθ = θ2 – θ1
Example problem #1

John Glenn, in 1962, circled the earth 3 times in
less than 5 hours. If his distance from the center
of the earth was 6560 km, what arc length did
he travel through? (answer in Km)
 G: Δ𝜃 = 3 𝑟𝑒𝑣 = 3(2𝜋 𝑟𝑎𝑑), 𝑟 = 6560 𝑘𝑚
 U: Δ𝑠 = ?
 E: ∆𝜃 = ∆𝑠𝑟
 S: ∆𝑠 = 𝑟∆𝜃
 S:

= (6560𝑘𝑚) (6𝜋)
= 𝟏. 𝟐𝟒 x 𝟏𝟎𝟓 𝒌𝒎
Example problem #2

While riding on a carousel that is rotating clockwise, a child travels
through an arc length of 11.5 m. If the child’s angular displacement is
165o,what is the radius of the carousel?

G: 𝑠
= −11.5𝑚, Δ𝜃 = −165𝑜;
Δ𝜃 (𝑟𝑎𝑑) =
=
U: 𝑟 = ?
∆𝑠
 E: ∆𝜃 = 𝑟
∆𝑠
 S: r = ∆𝜃 =


S: 3.99 m
−11.5
−2.88
𝜋
180
x Δθ(𝑑𝑒𝑔)
𝜋
𝑥 (−165) = −2.88 𝑟𝑎𝑑
180
Angular Displacement Example – Try this
Two people ride on a carousel. One rides on a
horse located 5 meters from the center. The
other rides on a swan located 3 meters from
the center.
 When the carousel goes around ¼ of a
revolution, how far does each person travel?

s  rDq
Where:
Horse: 7.85 m
Swan: 4.71 m
Dq 
2p
4
Angular speed (𝜔)



The rate at which a body rotates about an axis,
which is the rate of change of angular position.
expressed in: rad/sec or per sec. (T-1)
Average angular speed =
𝜔𝑎𝑣𝑔 =
∆𝜃
𝑡
𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑡𝑖𝑚𝑒
Example problem #3

In 1975, an ultra-fast centrifuge attained an average
angular speed of 2.65 x 104 rad/sec. What was the
angular displacement after 1.5 sec?
G: Ѡ = 2.65 x 104 𝑟𝑎𝑑𝑠
 U: Δθ = ?
∆𝜃
 E: 𝜔𝑎𝑣𝑔 =
𝑡


S: Δθ = Ѡavg t
= 2.65 x 104 x 1.5

S: 3.98 x 104 radians
𝑠𝑒𝑐 , t
= 1.5 s
Example problem #4

A child at an ice cream parlor spins on a stool. The child
turns counterclockwise with an average angular speed of
4.0 rad/s. In what time interval will the child’s feet have
an angular displacement of 8.0π rad?
G: Ѡ = 4.0 rads/s, Δθ = 8.0 π rad
 U: 𝑡 = ?


E: 𝜔𝑎𝑣𝑔 =

S: t = 𝜔

S: 𝟔. 𝟐𝟖 𝒔
∆𝜃
𝑎𝑣𝑔
∆𝜃
𝑡
=
8.0π 𝑟𝑎𝑑
4.0 𝑟𝑎𝑑/𝑠
=2.0 π s
Angular acceleration (α)



The rate of change of angular speed, increase or
decrease in rotational speed of the particle,
expressed in rad/s/s or T-2.
𝛼=
𝜔 𝑓 − 𝜔𝑜
𝑡
or
Dw
a
Dt
Angular acceleration =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑠𝑝𝑒𝑒𝑑
𝑡𝑖𝑚𝑒
◦ wf  final angular speed
◦ wo  initial angular speed
◦ a  angular acceleration
◦ Dt = elapsed time
Example problem #5

A car’s tire rotates at an initial angular speed of 21.5
rad/s. The driver accelerates, and after 3.5 s the tire’s
angular speed is 28.0 rad/s. What is the tire’s average
angular acceleration during the 3.5 s time interval?
G: Ѡo = 21.5 rad/s, Ѡf = 28.0 rad/s, t = 3.5s
 U: α = ?
𝜔𝑓 − 𝜔𝑜
 E: 𝛼 =
𝑡

28 −21.5
3.5

S: 𝛼 =

S: 1.86 rad/s2
In comparing angular and linear
quantities, they are similar. you may write this at
the bottom of your notes you will need it
Analogies Between Linear and
Rotational Motion:
DUE TOMORROW
SECTION
ASSIGNMENT
Tangential and centripetal acceleration
SECTION 2:
Tangent
line
Tangential speed
Objects in circular motion have a tangential speed
Tangent – a line that lies in a plane of circle that
intersects the circle at one point.
 Tangential speed (vt) is the instantaneous linear
speed of an object directed along the tangent to
the object’s circular path.


◦ Tangential speed = (distance from the axis) X (angular speed)
◦ 𝑣𝑡 = 𝑟𝜔
 ω = instantaneous angular speed
 This equation is valid only when ω is measured in rad/s
Example problem #1

The radius of a CD is 0.06 m. If a microbe riding on the
disc’s rim has a tangential speed of 1.88 m/s, what is the
microbe’s angular speed?
G: 𝑟 = 0.06𝑚, 𝑣𝑡
 U: ω = ?
 E: 𝑣𝑡 = 𝑟𝜔

𝑣𝑡
𝑟
1.88
0.06

S: 𝜔 =

S: 31.33 rad/s
=
= 1.88 𝑚/𝑠
Rotation and Centripetal
Acceleration

If an object is rotating
about a fixed axis, even
at a constant speed,
every point in that
object is undergoing a
centripetal acceleration
as well.
Tangential acceleration

Tangential acceleration (at) is defined as the
instantaneous linear acceleration of an object
directed along the tangent to the objects circular
path

Tangential acceleration = (distance from the axis) x (angular acceleration)

𝑎𝑡 = 𝑟𝛼
◦ α = is the instantaneous angular acceleration
◦ In this equation, you must use the unit radians to be valid
Centripetal acceleration
◦ Centripetal acceleration can be
calculated using angular speed as
well:
 𝑎𝑐 = 𝑟𝜔2
 𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑎𝑥𝑖𝑠 x 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑠𝑝𝑒𝑒𝑑2

Ex.-swinging a stopper above your
head
Centripetal Acceleration in Angular
Form
2
t
v
aC 
r
2
( rw )
aC 
r
2
aC  w r
Total acceleration

The total acceleration is
the vector sum of
centripetal acceleration,
which points to the
center of the circle, and
tangential acceleration,
which is tangent to the
circle.
at
aC
atotal  a  a
2
C
2
t
Example problem #2
A spinning ride at a carnival has an angular
acceleration of 0.50 rad/s2. How far from the
center is a rider who has a tangential
acceleration of 3.3 m/s2?
 G: 𝛼 = 0.50 𝑟𝑎𝑑/𝑠2, 𝑎𝑡 = 3.3 𝑚/𝑠2
 U: r = ?
 E: 𝑎𝑡 = 𝑟𝛼

𝑎𝑡
𝛼

S: 𝑟 =

S: 6.6 m
=
3.3
0.5
Rolling Objects – center of mass

The center of mass of a rolling wheel will
have the same velocity and acceleration as
a point on the edge of the wheel.
Rolling Objects
sCM  Rq
vCM  Rw
aCM  Ra
SECTION ASSIGNMENT