Angular Velocity
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Transcript Angular Velocity
More Trigonometry!!
Section 4-2
Review
Angles
Standard Position
Coterminal Angles
Reference Angles
Converting from Degrees –
degrees, minutes, seconds (DMS)
Angle-
formed by rotating a ray
about its endpoint (vertex)
Terminal Side Ending position
Initial Side Starting position
Standard Position Initial side on positive x-axis
and the vertex is on the origin
An angle describes the amount and direction of rotation
120°
–210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
Coterminal Angles: Two angles with the same initial
and terminal sides
Find a positive coterminal angle to 20º
20 360 380
Find a negative coterminal angle to 20º 20 360 340
Types of questions you will be asked:
Identify a) ALL angles coterminal with 45º, then b) find one
positive coterminal angle and one negative coterminal angle.
a) 45º + 360k (where k is any given integer).
b) Some possible answers are 405º, 765º, - 315º, - 675º
Decimal Degrees (DD)
• Decimal degrees are similar to degrees/ minutes/seconds
(DMS) except that minutes and seconds are expressed as
decimal values.
• Decimal degrees make digital storage of coordinates
easier and computations faster.
60.34444 instead of 60°20'40"
Converting from DMS to DD
Express 365010as decimal degrees (DD)
To complete the calculation, remember that …
1 degree = 60 minutes
1 minute = 60 seconds
1° = 60
1 = 60
3600
So … 1 degree = _________seconds
THEREFORE …
Try this: Converting DMS to DD
degrees
seconds
60º20'40"
minutes
20 minutes.= 0.33333 (20/60)
40 seconds = 0.01111 (40/3600)
Add up the degrees to get an answer:
60º + 0.33333 + 0.01111=60.34444 DD
Converting from DD to DMS
Express 50.525 in degrees, minutes, seconds
To reverse the process, we multiply by 60 instead.
50º + .525(60)
50º + 31.5
50º + 31 + .5(60)
50 degrees, 31 minutes, 30 seconds
Homework
Page 238 # 2 - 16 evens
So, what exactly is a RADIAN?
Many math problems are more
easily handled when degrees are
converted to RADIANS.
5
For a visual depiction of a radian,
let’s look at a circle.
Definition: a radian
is an arc length of
one radius
So, how many radians are
there in a given circle?
What’s the connection
between degrees and
radians?
6
4
θ
3
a little extra
r
2
1
radian
360 2 r
360
180
r
57.3
2
180
We can use the two ratios
to convert between radians and
or
180
degrees.
Example: Change 330˚ to radians:
330
11
180
6
In most cases, radians are left in
terms of π
2 radians to degree measure.
3
2 180
120
3
Example: Convert
Two formulas to know:
1.
Arc Length of a circle: S = rθ (θ in radians)
Example: Given a central angle of 128 degrees, find the length of the intercepted arc
in a circle of radius 5 centimeters. Round to nearest tenth.
S = rθ
2.
5 128
180
11.2 cm
Area of a sector (slice of pie): A = ½ r2θ
(θ in radians)
Example: Find the area of a sector of the central angle measures
the radius of the circle is 16 inches. Round to nearest tenth.
A=½
r2θ
1
5
2
16
335.1in 2
2
6
radians and
Linear & Angular Velocity
Things that turn have both a linear velocity
and an angular velocity.
Things that Turn - Examples
tire on a car or bike
buckets on a waterwheel
teeth on a gear
can on a kitchen cabinet lazy susan
propeller on an airplane
horse on a Merry-Go-Round
fins on a fan or a windmill
earth on its axis
Linear & Angular Velocity - Examples
film on a projector or tape on a videotape
turntable in a microwave oven
blade on a lawnmower
Earth around the sun
seat on a Ferris wheel
rope around a pulley
a record on an old record player
drum/barrel in a clothes dryer
Things that Turn - Examples
lock on your locker
hands on a clock
roller brush on a vacuum cleaner
tops & gyroscopes & dradle
motor crankshaft
blades in a blender
roller skate wheels
Carnival rides: tilt-a-whirl, scrambler, etc.
weather vane washing machine agitator
Angular Velocity
Definition:
Angular Velocity (ω): the speed
at which an angle opens.
Remember: θ is in radians.
Ex. 6 rev/min, 360°/day, 2π rad/hour
t
Angular Velocity
Example: determine the angular velocity if 7.3 revolutions are completed in 9
seconds. Round to nearest tenth.
1 revolution is 2π radians … so we’re talking about…
7.3 2 14.6 radians
Let’s use the formula:
t
14.6
9.2 rad / sec
9sec
Angular Velocity
EXAMPLE 2: A carousel makes 2 5/8 rotations per minute. Determine the
angular velocity of a rider on the carousel in radians per second .
2 85 2.625revolutions
2.625revs 1min 2 radians
radians
0.275
1min
60sec revolution
sec
Linear Velocity
Definition:
Linear Velocity: the speed with which
An object revolves a fixed distance
from a central point.
If you already know the angular velocity, then
…
Ex. 55 mph, 6 ft/sec, 27 cm/min, 4.5 m/sec
vr
t
r
Linear Velocity
In the carousel scenario, one of the animals
is 20 feet from the center. What is its linear
velocity?
Solution
The cable moves at a fixed speed … a linear velocity.
r
t
.275radians
20
sec
5.5 ft/sec