2. WIND TURBINES such
as these can generate
significant energy in a
way that is environ-
mentally friendly and
renewable. The
concepts of rotational
acceleration, angular
velocity, angular
displacement, rotational
inertia, and other topics
discussed in this
chapter are useful in
describing the operation
of wind turbines.
3. Rotational Displacement,
Consider a disk that rotates from A to B:
A
B
Angular displacement :
Measured in revolutions,
degrees, or radians.
1 rev = 360 0 = 2 rad
The best measure for rotation of
rigid bodies is the radian.
4. Definition of the Radian
One radian is the angle subtended at
the center of a circle by an arc length s
equal to the radius R of the circle.
1 rad = = 57.30
R
R
s
s
R
5. Example 1: A rope is wrapped many times
around a drum of radius 50 cm. How many
revolutions of the drum are required to
raise a bucket to a height of 20 m?
h = 20 m
R
= 40 rad
Now, 1 rev = 2 rad
= 6.37 rev
1 rev
40 rad
2 rad
20 m
0.50 m
s
R
6. Example 2: A bicycle tire has a radius of
25 cm. If the wheel makes 400 rev, how
far will the bike have traveled?
= 2513 rad
s = R = 2513 rad (0.25 m)
s = 628 m
2 rad
400 rev
1 rev
7. Angular Velocity
Angular velocity,w, is the rate of change in
angular displacement. (radians per second.)
w 2f Angular frequency f (rev/s).
Angular velocity can also be given as the
frequency of revolution, f (rev/s or rpm):
w Angular velocity in rad/s.
t
8. Example 3: A rope is wrapped many times
around a drum of radius 20 cm. What is
the angular velocity of the drum if it lifts the
bucket to 10 m in 5 s?
h = 10 m
R
w = 10.0 rad/s
w
t
50 rad
5 s
= 50 rad
10 m
0.20 m
s
R
9. Example 4: In the previous example, what
is the frequency of revolution for the drum?
Recall that w = 10.0 rad/s.
h = 10 m
R
f = 95.4 rpm
2 or
2
f f
w
w
10.0 rad/s
1.59 rev/s
2 rad/rev
f
Or, since 60 s = 1 min:
rev 60 s rev
1.59 95.5
1 min min
f
s
10. Angular Acceleration
Angular acceleration is the rate of change in
angular velocity. (Radians per sec per sec.)
The angular acceleration can also be found
from the change in frequency, as follows:
2 ( )
2
f
Since f
t
w
2
Angular acceleration (rad/s )
t
w
11. 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. What is the
average angular acceleration?
h = 20 m
R
= 4.00 rad/s2
0
f o f
or
t t
w w w
2
16 rad/s rad
4.00
4 s s
12. Angular and Linear Speed
From the definition of angular displacement:
s = R Linear vs. angular displacement
v = w R
s R
v R
t t t
Linear speed = angular speed x radius
13. Angular and Linear Acceleration:
From the velocity relationship we have:
v = wR Linear vs. angular velocity
a = R
v v R v
v R
t t t
Linear accel. = angular accel. x radius
14. Examples:
R1 = 20 cm
R2 = 40 cm
R1
R2
A
B
wo = 0; wf = 20 rad/s
t = 4 s
What is final linear speed
at points A and B?
Consider flat rotating disk:
vAf = wAf R1 = (20 rad/s)(0.2 m); vAf = 4 m/s
vBf = wBf R1 = (20 rad/s)(0.4 m); vBf = 8 m/s
15. Acceleration Example
R1 = 20 cm
R2 = 40 cm
What is the average angular
and linear acceleration at B?
R1
R2
A
B
wo = 0; wf = 20 rad/s
t = 4 s
Consider flat rotating disk:
= 5.00 rad/s2
a = R = (5 rad/s2)(0.4 m) a = 2.00 m/s2
0 20 rad/s
4 s
f
t
w w
16. Angular vs. Linear Parameters
Angular acceleration is the time
rate of change in angular velocity.
0
f
t
w w
Recall the definition of linear
acceleration a from kinematics.
0
f
v v
a
t
But, a = R and v = wR, so that we may write:
0
f
v v
a
t
becomes
0
f
R R
R
t
w w
17. A Comparison: Linear vs. Angular
0
2
f
v v
s vt t
0
2
f
t t
w w
w
f o t
w w
f o
v v at
2
1
0 2
t t
w
2
1
0 2
s v t at
2
1
2
f t t
w
2 2
0
2 f
w w
2 2
0
2 f
as v v
2
1
2
f
s v t at
18. Linear Example: A car traveling initially
at 20 m/s comes to a stop in a distance
of 100 m. What was the acceleration?
100 m
vo = 20 m/s vf = 0 m/s
Select Equation:
2 2
0
2 f
as v v
a = =
0 - vo
2
2s
-(20 m/s)2
2(100 m)
a = -2.00 m/s2
19. Angular analogy: A disk (R = 50 cm),
rotating at 600 rev/min comes to a stop
after making 50 rev. What is the
acceleration?
Select Equation:
2 2
0
2 f
w w
= =
0 - wo
2
2
-(62.8 rad/s)2
2(314 rad)
= -6.28 m/s2
R
wo = 600 rpm
wf = 0 rpm
= 50 rev
2 rad 1 min
600 62.8 rad/s
min 1 rev 60 s
rev
50 rev = 314 rad
20. Problem Solving Strategy:
Draw and label sketch of problem.
Indicate + direction of rotation.
List givens and state what is to be found.
Given: ____, _____, _____ (,wo,wf,,t)
Find: ____, _____
Select equation containing one and not
the other of the unknown quantities, and
solve for the unknown.
21. Example 6: A drum is rotating clockwise
initially at 100 rpm and undergoes a constant
counterclockwise acceleration of 3 rad/s2 for
2 s. What is the angular displacement?
= -15.0 rad
Given: wo = -100 rpm; t = 2 s
= +3 rad/s2
2 2
1 1
2 2
( 10.5)(2) (3)(2)
ot t
w
rev 1 min 2 rad
100 10.5 rad/s
min 60 s 1 rev
= -21.0 rad + 6 rad
Net displacement is clockwise (-)
R
