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In the picture are three gears.
Each gears rotation causes
rotation of another.
Suppose the radius of purple, yellow and red gears
are x, 2x, and 3x respectively, by how much faster
or slower does the purple gear rotate comparedto red?
Rotational Motion
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Rotational, Linear & Angular Velocity
If the disc will make one complete rotation in 1 second,
which point (A, B, or C) moves the fastest?
x
A
2x
B
3x
C
Rotational, Linear & Angular Velocity
When the disc makes a complete rotation, how are you
going to determine the distance it covers?
How are the distances covered by A compared to that
of B and C?
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Rotational, Linear & Angular Velocity
When the disc make a quarter turn from Position 1 to Position 2,
points A, B and C will also move and align to the final position.
If the displacement (d) is extended into straight path, the lineardisplacement of C is obviously the longer. While moving at equal
time with others, it is moving the fastest. That speed is called
linear speed (v).
1
2
Rotational, Linear & Angular Velocity
In some respect, the three points move together at the same
time. The change of position can be reflected by the change in
angle--- which the three show the same change.
The change in angle in a given amount of time is calledangular
speed ( )
1
2
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Rotational, Linear & Angular Velocity
Notice that the angular speedof the three are the same. But the
distance covered by point C is the greatest. Notice also the
distance of C from the axis of rotation (radius), it is the farthest.Therefore, the farther is the object from the center, the bigger is
the distance (d) covered.
1
2
In the picture are three gears.
Each gears rotation causes
rotation of another.
Suppose the radius of purple, yellow and red gears
are x, 2x, and 3x respectively, by how much faster
or slower does the purple gear rotate compared
to red?
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Rotational, Linear & Angular Velocity
Consider a disc with a radius of 2 meters. The length of its edge(circumference) is measured by the counter. When one complete
revolution has elapsed, what reading will be registered in the
counter?
r=2m 0.0m
Rotational, Linear & Angular Velocity
When the disc rolls on the ground, one complete revolution of it
will cover a distance similar to its diameter.
How far does a disc of diameter 2m cover after rolling at a speed
of 60rpm in 1 minute?
r=2m 0.0m
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Rotational, Linear & Angular Velocity
Getting the relationship of all:
S r
t
Sv
t
S
v
r
r
t t
S rt
Rotational, Linear & Angular Velocity (Problems)A pair of identical 1000-kg space pods in outer space
are connected to each other by a 900-m-long cable.
They rotate about a common point like a spinning
dumbbell at 1.2 rpm. What is its angular velocity?
(rad/s)
.
t
rev
1 2
min rev
2 min1
.
s
rad
s
60
0 13
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Rotational, Linear & Angular Velocity (Problems)
A calesa with wheels whose diameter is 1.5 meters is
traveling at 24 kph. Find the angular velocity of thewheel in revolutions per minute.
Pretend that it only travels for one hour. Then calesa would have traveled
24km or 24000m.
When the calesa makes one rotation it would travel the distance of its
circumference.
S= r
or
= D
= 1.5
=4.71m
r
2
S is the circumference
D is the diameter
This is the distance traveled by
the calesa wheel per rotation.
Rotational, Linear & Angular Velocity (Problems)A calesa with wheels whose diameter is 1.5 meters is
traveling at 24 kph. Find the angular velocity of the
wheel in revolutions per minute.
total distance traveled
no. of rotations= distance covered per rotation
.
. rotations
m
rotation
m24000
4 71
5095 5
Pretend that it only travels for one hour. Then calesa would have traveled
24km or 24000m.
When the calesa makes one rotation it would travel the distance of its
circumference.
This is the total number of
rotations made by calesa
wheel in 1 hour or 60 minutes.
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Rotational, Linear & Angular Velocity (Problems)
A calesa with wheels whose diameter is 1.5 meters is
traveling at 24 kph. Find the angular velocity of thewheel in revolutions per minute.
5095.5 rev 1hour=
hour 60min
.min
rev
84 9
Pretend that it only travels for one hour. Then calesa would have traveled
24km or 2400m.
When the calesa makes one rotation it would travel the distance of its
circumference.
The wheel of diameter 1.5m
can make this much of
rotation every minute orrpm.
Angular Acceleration
When a disc begins to rotate, its initial rotational velocity is zero,therefore, its angular velocity is also zero.
When it rotates, it will gain rotational speed at increasing rate
until it reaches its desired rpm.
During this time of gaining rotational speed, the angular speed
also changes. Therefore we say that it has angular acceleration.
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Angular Acceleration
Did you notice that when the angle changes rapidly, the linear
displacement also changes rapidly?
When the rotation is accelerating angularly, the movement is
also accelerating linearly.
vr
t t
a r
Angular AccelerationIn constant angular
acceleration
f i
t
In constant linear
acceleration
f iv v
at
f it
2
f iv v
d t2
itt
2
2i
atd v t
2
2
f i
2 2
2
f iv v
da
2 2
2
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Angular Acceleration (Problems)
When a record player is stopped, the turn table, initially
rotating at 33.3rpm, slows down and stop in 60s.Calculate the angular acceleration of the turn table.
=t
.=
60s
.
f i
rad
s
rad
s
2
0 3 49
0 058
. min=
min
.=
60s
.
rev
rev s
rad
rad
s
33 3 2 1
60
209 23
3 49
Angular Acceleration (Problem)When the rotational speed of a rotating object changes
from 2rpm to 3rpm in 30s, how much does it
accelerate? Express your answer in (rad/s2).
=
=
t
rev
1
min
. rad
rev
6 28
1
min1
.
ss
rad
s2
60
30
0 00349
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Angular Acceleration (Problem)
A Ferris wheel starts from rest and acquires an angular
velocity of 0.5rad/s after completing one revolution.Calculate the distance covered by the person seated
20m from the axis of rotation.
.
.
S r
r
m
m
2
6 28 20
125 6
Rotational Inertia
A B
Which is easier to balance? Why?
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Rotational Inertia
As linear inertia is a bodys resistance to change in motion,
rotational inertia is the bodys resistance to change in rotation.
In linear inertia, change in motion is caused by a net force
In rotational inertia, change in rotation is caused by a net torque.
Linear inertia is bigger when mass is bigger.
Rotational inertia is bigger when the distance between center of
mass and axis of rotation is bigger.
Rotational Inertia
Explain why a barbell with
weights near the hand easy
to rotate that the one with
weights far from the hand?
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Rotational Inertia
How can a long pole help atight rope walker?
Rotational Inertia
Which disc will roll faster?
Why? A B
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Angular Momentum
A mass that moves in a straight path has linear momentum that can
derived as a product oflinear inertia (mass) and linear velocity.
p=mv
A mass that rotates has angular momentum which can be derived as
a product ofrotational inertia and angular velocity.
L=I
Angular Momentum (Problem)
A mass that moves in a straight path has linear momentum that can
derived as a product oflinear inertia (mass) and linear velocity.
p=mv
A mass that rotates has angular momentum which can be derived asa product ofrotational inertia and angular velocity.
L=I
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Conservation of Angular Momentum (Problem)
A pair of identical 1000-kg space pods in outer space
are connected to each other by a 900-m-long cable.They rotate about a common point like a spinning
dumbbell at 1.2 rpm. If one of the pods pulls in 100m
of cable what will be the systems new rotation rate?bef ore afte r
b ef or e b ef or e a ft er a ft er
L L
I I
m
before beforer m
2
.
.
after after
before before
after
after
r
r
r
m rpm
m
rpm
2
2
2
2
2
450 1 2
400
1 5
m recurs in both sides of the equation so it can be
omitted.
When the diameter is reduced by 100m,
the radius will be reduced by only 50m.
If there are errors in the presentation, please feel free to
comment.