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Unit 7 – Angular Motion
WIND TURBINES such
as these can generate
significant energy in a
way that is environmentally friendly and
renewable
...
Objectives: After completing this
module, you should be able to:
• Define and apply concepts of angular
displacement, velocity, and acceleration
...
• Write and apply relationships between
linear and angular parameters
...
• Apply the following concepts to rotation:
1
...
Rotational kinetic energy and
momentum
3
...
1 rev = 360 0 = 2 rad
The best measure for rotation of
rigid bodies is the radian
...
s
s
R
1 rad =
R
R
= 57
...
How many
revolutions of the drum are required to
raise a bucket to a height of 20 m?
s
20 m
R 0
...
37 rev
h = 20 m
Example 2: A bicycle tire has a radius of
25 cm
...
25 m)
s = 628 m
Angular Velocity
Angular velocity,, is the rate of change in
angular displacement
...
)
t
Angular velocity in rad/s
...
Example 3: A rope is wrapped many times
around a drum of radius 20 cm
...
20 m
t
= 50 rad
R
50 rad
5s
h = 10 m
= 10
...
0 rad/s
...
0 rad/s
f
1
...
59
95
...
5 rpm
h = 10 m
Angular Acceleration
Angular acceleration is the rate of change in
angular velocity
...
)
t
Angular acceleration (rad/s 2 )
The angular acceleration can also be found
from the change in frequency, as follows:
2 (f )
t
Since
2 f
Example 5: The block is lifted from rest
until the angular velocity of the drum is
16 rad/s after a time of 4 s
...
00 2
4s
s
h = 20 m
= 4
...
angular displacement
s R
v
t t t
R
v=R
Linear speed = angular speed x radius
Angular and Linear Acceleration:
From the velocity relationship we have:
v = R Linear vs
...
= angular accel
...
2 m);
vAf = 4 m/s
vAf = Bf R1 = (20 rad/s)(0
...
4 m)
R2
R1 = 20 cm
R2 = 40 cm
= 5
...
00 m/s2
Angular vs
...
a
v f v0
t
But, a = R and v = R, so that we may write:
a
v f v0
t
becomes
R
Angular acceleration is the time
rate of change in angular velocity
...
Angular
v0 v f
s vt
2
t
0 f
t
2
v f vo at
f o t
s v0t at
1
2
2
0t t
1
2
2
2
f t t
2
2
0
2
2
0
s v f t at
1
2
2as v v
2
f
t
1
2
2
f
Linear Example: A car traveling initially
at 20 m/s comes to a stop in a distance
of 100 m
...
00 m/s2
Angular analogy: A disk (R = 50 cm),
rotating at 600 rev/min comes to a stop
after making 50 rev
...
8 rad/s
min 1 rev 60 s
=
0 - o2
2
=
-(62
...
29 rad/s2
Problem Solving Strategy:
Draw and label sketch of problem
...
List givens and state what is to be found
...
Example 6: A drum is rotating clockwise
initially at 100 rpm and undergoes a constant
counterclockwise acceleration of 3 rad/s2 for
2 s
...
5 rad/s
min 60 s 1 rev
ot t (10
...
9 rad + 6 rad
1
2
R
2
= -14