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0
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Activity: Think-Pair-Share
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0
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With your partner, in a ½ sheet of paper, list at
least 2 types of rotating objects and how are they
important to society. Write your answer in tabular
form below. Afterwards, volunteered pairs will
share their output in class.
Types of Rotating
Objects
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40
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60
40
20
0
Importance to
Society
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40
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60
80
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General Physics 1
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80
60
60
Rotational
Kinematics
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0
20
Lesson 1
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General Physics 1
Lesson Objectives
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80
Define kinematic rotational variables
01 such as angular position, angular
velocity, and angular acceleration.
02 Derive rotational kinematic equations.
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40
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0
20
60
40
20
0
20
Solve for the angular position, angular
03 velocity, and angular acceleration of a
rotating body.
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0
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0
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180
❑ Kinematics – studies the description of motion
80
80
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0
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60
80
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of objects.
❑ Rotational Kinematics –it’s all
about any object that can
rotate or spin.
❑ The kinematics of rotational
motion describes the
relationships among rotation
angle (θ), angular velocity (ω),
angular acceleration (α), and
time (t).
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0
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Consider a thin disk of radius r spinning on its axis
80
80
❑This disk is a real object,
it has structure.
❑We call these kinds of
objects rigid bodies.
❑Rigid bodies do not
bend twist, or flex; for
example, a billiard ball.
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0
20
40
60
80
In the simplest kind of rotation, points on a rigid
object move on circular paths around an axis of
rotation.
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0
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80
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40
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0
0
20
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0
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20
0
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20
0
0
20
20
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40
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120
100
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40
20
0
20
40
60
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100
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180
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160
140
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100
80
60
40
20
0
20
Angular Displacement
40
60
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100
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80
60
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40
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20
0
0
20
20
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180
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140
120
100
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60
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20
0
20
40
60
80
100
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140
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100
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60
40
20
0
20
Angular Displacement
40
60
80
100
120
140
160
180
80
80
❑The angle of rotation, θ, is
the angular displacement,
measured in radians.
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40
20
0
60
40
20
0
❑For one complete
revolution, conversion
relation:
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20
40
40
60
60
80
80
𝟐𝝅 𝒓𝒂𝒅 = 𝟑𝟔𝟎° = 𝟏 𝒓𝒆𝒗
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100
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0
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100
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20
Sample Problems
0
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40
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80
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180
80
80
1) An object travels around a circle 10 full
60
60
turns in 2.5 seconds. Calculate (a) the
angular displacement, θ, in radians.
2) A girl goes around a circular track that
has a diameter of 12 m. If she runs
around the entire track for a distance of
100
m,
what
is
her
angular
displacement?
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0
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80
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140
120
100
80
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40
20
0
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60
80
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140
160
40
20
0
20
40
60
80
180
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100
80
60
40
20
Sample Problems
0
20
40
60
80
100
120
140
160
180
80
80
3) Synchronous
satellites
are put into an orbit
whose
radius
is
4.23×107m. If the angular
separation of the two
satellites is 2.00 degrees,
find the arc length (in
km)
that
separates
them.
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40
20
0
20
40
60
80
180
160
140
120
100
80
60
40
20
0
20
40
60
40
20
0
20
40
60
80
60
80
100
120
140
160
180
180
160
140
120
100
80
60
40
Angular Velocity
80
60
40
20
0
20
40
60
80
100
120
140
160
180
80
❑ In rotational motion, angular velocity (ω) is
defined as the change in angular displacement
(θ) per unit of time (t). In symbol,
60
40
20
20
𝝎=
0
20
40
60
∆𝜽
∆𝒕
0
20
❑ Unit is as revolution per second
(rev/sec, rps), or radian per second
40
60
𝝎 = 𝟐𝝅𝒇
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180
160
140
120
100
80
60
40
20
80
0
20
40
60
80
100
120
140
160
180
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140
120
100
80
60
40
Angular Velocity
20
0
20
40
60
80
100
120
140
160
180
80
80
❑ From linear velocity conversion, we have
60
𝝎=
40
20
𝒗𝒕
𝒓
60
𝒗𝒕 = 𝒓𝝎
OR
40
20
where:
ω is the angular velocity (rad/s),
𝑣𝑡 is the tangential velocity (m/s), and
r is the radius in circular path (meters).
❑ The farther away the object is from the center of
rotation, the higher the tangential velocity.
0
0
20
20
40
40
60
60
80
80
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140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
80
80
60
60
40
40
20
20
0
0
20
20
40
40
60
60
80
80
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
180
160
140
120
100
80
60
40
20
Sample Problems
0
20
40
60
80
100
120
140
160
180
80
80
60
60
1) If an object travels around a circle
40
40
with an angular displacement of
70.8 radians in 3.0 seconds, what
is its average angular velocity ω in
(rad/s)?
20
0
20
40
20
0
20
40
60
60
80
80
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
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140
120
100
80
60
40
20
Sample Problems
0
20
40
60
80
100
120
140
160
180
80
80
2) A gymnast on a
60
40
20
0
20
40
60
80
180
60
high bar swings
through two
revolutions in a
time of 1.90 s. Find
the average
angular velocity
(in rad/s) of the
gymnast.
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140
120
100
80
60
40
20
40
20
0
20
40
60
80
0
20
40
60
80
100
120
140
160
180
Angular Acceleration
180
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100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
❑ Changing angular velocity means that an angular
acceleration is occurring over time.
80
80
60
60
❑ The angular acceleration, α, of a rotating object is
the rate at which the angular velocity changes
with respect to time.
40
40
20
20
0
0
20
20
∆𝝎 𝝎𝒇 − 𝝎𝒊
𝜶=
=
∆𝒕
𝒕𝒇 − 𝒕𝒊
40
60
40
60
❑ Unit of measure is radian per second squared
(rad/s2).
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180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
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160
180
Angular Acceleration
180
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100
80
60
40
20
0
20
40
60
80
100
120
140
❑ If there is angular acceleration,
there will also be tangential
acceleration, since the tangential
velocity is changing.
160
180
80
80
ac
60
60
40
40
20
20
𝒗𝒕 𝟐
𝜶𝒕 = 𝒓𝜶 and 𝜶𝒄 =
= 𝝎𝟐 𝒓
𝒓
❑We can find a total resultant
acceleration 𝜶, since 𝒂𝒕 and 𝒂𝒄 are
perpendicular,
0
0
20
20
40
40
60
60
𝑎𝑐
80
𝒂=
180
160
𝟐
𝒂𝒄 + 𝒂𝒕
140
120
100
𝟐
80
60
∅ = 𝒕𝒂𝒏
40
20
0
−𝟏
(𝒂𝒕 /𝒂𝒄 )
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40
60
80
100
120
140
160
180
80
180
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140
120
100
80
60
40
80
60
40
20
0
20
40
60
80
100
120
140
160
180
❑ Although every particle in the
object has the same angular
ac
acceleration,
its
tangential
acceleration differs proportional
to its distance from the axis of
rotation.
❑The father away the particle is
from the rotation axis, the more
radial/centripetal acceleration it
receives.
80
𝜶𝒕 = 𝒓𝜶
60
40
20
20
0
0
𝟐
𝒗𝒕
𝜶𝒄 =
𝒓
𝟐
𝜶𝒄 = 𝝎 𝒓
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40
60
80
180
160
140
120
100
20
40
60
80
80
60
40
20
0
20
40
60
80
100
120
140
160
180
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
❑Angular displacement, angular
velocity, and angular acceleration
are all vector quantities.
80
80
60
60
40
40
20
20
0
❑The direction for angular velocity
and angular acceleration can be
determined by the right-hand rule.
0
20
20
40
40
60
60
80
80
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
Right-Hand Rule
180
80
60
40
20
0
20
40
60
80
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
A
convenient
tool
in
determining the direction of the
angular velocity. Defining the zaxis as the axis of rotation, curl
your fingers of your right hand in
the direction of rotation, your
thumb points in the direction of
the
angular
velocity
and
acceleration.
180
160
CCW is +, while CW is –
140
120
100
80
60
40
20
0
20
40
80
60
40
20
0
20
40
60
80
60
80
100
120
140
160
180
180
160
140
120
100
80
60
40
20
Sample Problems
0
20
40
60
80
100
120
140
160
180
80
80
1) A
60
40
jet awaiting clearance for takeoff has
momentarily stopped on the runway. As seen
from the front of one engine, the fan blades are
rotating with an angular velocity of –110 rad/s,
where the negative sign indicates a clockwise
rotation. 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.
60
40
20
20
0
0
20
20
40
40
60
60
80
80
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
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140
120
100
80
60
40
20
Sample Problems
0
20
40
60
80
100
120
140
160
180
2) A helicopter blade has an
80
60
40
20
0
20
40
60
80
180
angular speed of 6.50 rev/s
and an angular acceleration
of 1.30 rev/s2. For point 1 on
the
blade,
find
the
magnitude
of
(a)
the
tangential speed, (b) the
tangential acceleration, (c)
centripetal
acceleration,
and
the
(d)
resultant
acceleration.
160
140
120
100
80
60
40
20
0
20
40
80
60
40
20
0
20
40
60
80
60
80
100
120
140
160
180
Variables for Rotational and Translational
Kinematics
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
80
80
60
60
40
❑Consider a correlation of variables below:
20
Translational
d
displacement
v
velocity
acceleration
𝑎
t
time
0
20
40
60
80
180
160
140
120
100
80
60
40
20
0
20
40
40
20
Rotational
𝜽
𝝎
𝜶
t
60
80
100
120
140
160
0
20
40
60
80
180
Kinematic Equations for Rotational and
Translational Motion
180
80
160
60
40
20
0
20
40
60
80
180
160
140
120
100
80
60
40
Rotational
(𝛼 is constant)
𝝎𝒊 + 𝝎𝒇
𝜽=(
)𝒕
𝟐
𝝎𝒇 = 𝝎𝒊 + 𝜶𝒕
𝟏 𝟐
𝜽 = 𝝎𝒊 𝒕 + 𝜶𝒕
𝟐
𝝎𝒇 𝟐 = 𝝎𝒊 𝟐 + 𝟐𝜶𝜽
140
120
100
80
60
40
20
20
0
0
20
20
40
60
80
100
120
140
60
80
100
120
180
80
Translational
(a is constant)
𝒗𝒊 + 𝒗𝒇
𝒅=(
)𝒕
𝟐
𝒗𝒇 = 𝒗𝒊 + 𝒂𝒕
𝟏 𝟐
𝒅 = 𝒗𝒊 𝒕 + 𝒂𝒕
𝟐
𝒗𝒇 𝟐 = 𝒗𝒊 𝟐 + 𝟐𝒂𝒅
40
160
140
60
40
20
0
20
40
60
80
160
180
Tips for Problem Solving
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
80
80
❑Similar to the techniques used in linear
motion problems
o With constant angular acceleration, the
techniques are much like those with
constant linear acceleration
❑There are some differences to keep in mind
o For rotational motion, define a rotational
axis
▪ The choice is arbitrary
60
60
40
40
20
20
0
0
20
20
40
40
60
60
80
80
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
Tips for Problem Solving
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
80
80
❑There are some differences to keep in mind
▪ Once you make the choice, it must be
maintained
▪ In some problems, the physical
situation may suggest a natural axis
o The object keeps returning to its original
orientation, so you can find the number of
revolutions made by the body.
60
60
40
40
20
20
0
0
20
20
40
40
60
60
80
80
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
Kinematic Equations for Rotational and
Translational Motion
180
80
160
60
40
20
0
20
40
60
80
180
160
140
120
100
80
60
40
Rotational
(𝛼 is constant)
𝝎𝒊 + 𝝎𝒇
𝜽=(
)𝒕
𝟐
𝝎𝒇 = 𝝎𝒊 + 𝜶𝒕
𝟏 𝟐
𝜽 = 𝝎𝒊 𝒕 + 𝜶𝒕
𝟐
𝝎𝒇 𝟐 = 𝝎𝒊 𝟐 + 𝟐𝜶𝜽
140
120
100
80
60
40
20
20
0
0
20
20
40
60
80
100
120
140
60
80
100
120
180
80
Translational
(a is constant)
𝒗𝒊 + 𝒗𝒇
𝒅=(
)𝒕
𝟐
𝒗𝒇 = 𝒗𝒊 + 𝒂𝒕
𝟏 𝟐
𝒅 = 𝒗𝒊 𝒕 + 𝒂𝒕
𝟐
𝒗𝒇 𝟐 = 𝒗𝒊 𝟐 + 𝟐𝒂𝒅
40
160
140
60
40
20
0
20
40
60
80
160
180
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
Sample Problem: Blending with a Blender
180
80
80
The blades of an electric blender
are whirling with an angular velocity of
+375 rad/s while the “puree” button is
pushed in. When the “blend” button is
pressed, the blades accelerate and
reach a greater angular velocity after
the blades have rotated through an
angular displacement of +44.0 rad
(seven
revolutions).
The
angular
acceleration has a constant value of
+1740 rad/s2. Find the final angular
velocity of the blades.
60
60
40
40
20
20
0
0
20
20
40
40
60
60
80
80
180
160
140
120
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180
180
160
140
120
100
80
Group Task
80
60
40
20
0
20
40
60
80
100
120
140
160
180
80
A wheel rotates with a constant
angular acceleration of 3.5 rad/s2. If the
angular speed of the wheel is 2.0 rad/s at
t=0
a) through what angle does the wheel
rotate between t = 0 and t = 2.0 s? Give
your answer in radians and in
revolutions.
b)What is the angular speed of the wheel
at t = 2.0 s?
60
40
20
0
20
40
60
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Group Task
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A figure skater is spinning with an
angular velocity of +15 rad/s. She then
comes to a stop over a brief period of
time. During this time, her angular
displacement is +5.1 rad. Determine
a) her average angular acceleration
b) the time during which she comes to
rest.
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Group Task
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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 magnitude of the
total acceleration.
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Group Task
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A centrifuge in a medical
laboratory is rotating at an angular
speed of 3600 rev/min. When
switched off, it rotates 50 times
before coming to rest. Find the
constant angular deceleration of the
centrifuge.
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Group Task
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A grindstone wheel has a
constant angular acceleration of 0.35
rad/s2.
a) If it starts from rest, what is the
angular displacement after 18 s?
b) If it starts from rest, what is the
angular speed of the wheel at t =
18 s?
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Challenge Questions:
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1) Angular acceleration does not change with
radius, but tangential acceleration does. True
or False? Why?
True. The angular acceleration of an object is
the same because the entire object moves as a
rigid body through the same angle in the same
amount of time. While, tangential acceleration
has this concept,
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Challenge Questions:
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2) Differentiate angular acceleration from
tangential (or linear) acceleration.
Angular acceleration is the change in
angular velocity divided by time, while
tangential acceleration is the change in
linear velocity divided by time.
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Challenge Questions:
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3) On a rotating carousel or merry-go-round, one child
sits on a horse near the outer edge and another
child sits on a lion halfway out from the center.
Which child has the greater linear velocity? Which
child has the greater angular velocity?
Linear velocity is the distance traveled divided by
the time interval. So, the child sitting at the outer edge
travels more distance within the given time than the
child sitting closer to the center. Thus, the child on the
horse has greater linear velocity.
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Challenge Questions:
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3) On a rotating carousel or merry-go-round, one child
sits on a horse near the outer edge and another
child sits on a lion halfway out from the center.
Which child has the greater linear velocity? Which
child has the greater angular velocity?
Angular velocity is the angle traveled divided by
the time interval. The angle both the children travel in
the given time interval is the same. Thus, both the
child on the horse and that on the lion have the same
angular velocity.
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Match Me Right
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Match column A with column
B according to their meaning.
Write only the letter of your answer
on a ¼ sheet of paper.
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2.
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5.
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Column A (Meaning/Definition)
A measure of how angular velocity
changes over time.
The imaginary or actual axis around
which an object may rotate.
It is the change in linear velocity
divided by time.
It is half of the circle’s circumference.
The orientation of a body or figure
with respect to a specified reference
position as expressed by the amount
of rotation necessary to change from
one orientation to the other about a
specified axis.
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Column B (Terms)
A. Angular
position
B. Linear velocity
C. Axis of rotation
D. Tangential
acceleration
E. Angular
velocity
F. Kinematics
G. Angular
acceleration
H. Radian
I. Angular
displacement
J. Radius
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Column A (Meaning/Definition)
The rate of rotation around an axis is
usually expressed in radian or
revolutions per second or per
minute.
Branch of dynamics that deals with
aspects of motion apart from
considerations of mass and force.
It is the rate of change of the position
of an object that is traveling along a
straight path.
It is an angle whose corresponding
arc in a circle is equal to the radius of
the circle.
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Column B (Terms)
A. Angular
position
B. Linear velocity
C. Axis of rotation
D. Tangential
acceleration
E. Angular
velocity
F. Kinematics
G. Angular
acceleration
H. Radian
I. Angular
displacement
J. Radius
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Solve the following problem. Show your
solution.
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What is the angular acceleration
of a ball that starts at rest and
increases
its
angular
velocity
uniformly to 5 rad/s in 10 s?
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Agreement:
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Research about torque and
its formula. Also, research about
the conditions of a system under
static equilibrium.
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