3. Intercepted Arc
- arc length formed by the rays of
the central angle and the section of
the circle.
4. - It is the rate of movement of a
particle moving in constant speed
along a circular arc of radius r.
- It is given by ratio of the arc
length over time.
Linear Speed
5. – the rate of movement in
which the radius r forms an angle to
correspond with the arc length s.
Angular Speed
6. The Length of a Circular Arc
• The lengths of the arc intercepted by the
central angle is given by:
Where:
r is the radius of the circle
θ is the non negative radian measure
of a central angle of the circle.
rs
7. Examples
• A circle has a radius of 20
centimeters. Find the length of the
arc intercepted by a central angle of
120°.
Answer:
41.89 cm.
8. • The intercepted arc of an angle that
measures 120° is equal to 42
centimeters. Find the measure of the
radius of the circle.
Examples
9. • The length of the intercepted arc of a
circle with a radius of 20 centimeters
is 41.89 centimeters. Find the
measure of the intercepted angle.
Examples
10. Linear Speed
the linear speed of the particle is
given by:
Where:
s is the length of the arc traveled
t is the length of time
t
s
v
11. Example
• The second hand of a clock is 10.5
centimeters long. Find the linear
speed of the tip of this second hand
as it passes around the clock face.
12. Angular Speed
The angular speed of the particle is
given by:
Where:
θ is the angle in radians corresponding
to an arc length s,
t is the length of time
t
13. • Consider the motion of a point P along a
circle of radius 5 cm. Starting at A(5,0).
P
0 A(5,0)
14. 1. Suppose that OP after 10 seconds, the
angle covered by OP is .radians
4
5
Solve for the angular speed in radians
per second.
Answer: secondperradians
8
15. Linear Speed In Terms Of Angular
Speed
Formula:
Where:
r is the radius
w is the angular speed
rv
16. Examples
1. A wheel with a diameter 12 inches
is rotating at 340 revolutions per
minute. Find the angular speed in
radians per second and the linear
speed of a point in the wheel in
terms of meter per second.
17. • For angular speed:
• For linear speed:
3
34
secs.60
min.1
rev1
rad2
min.
rev.340
s
mm 4618515340
3
34
..
18. 2. A car with a tire with a radius of 16
inches is rotating at 450 revolutions per
minute.
a. the angular speed in radians per
second.
b. Linear speed of a point on the rim in
meters per second.
c. The measure α of the angle generated
by a spoke of the wheel in a 30
seconds.
19. Answer key for the examples:
a. ω = 3π/2
b. v = 1.9151 m/s
Note: get the linear speed with the use of the
angular speed and and the radius.
c. α = 45π rad
Note: Multiply the angular speed by the
length of time t to get the measure of the
angle formed.
20. Exercises
Give the radian measure of the central
angle of a circle that intercepts an
arc of length s.
• r = 16 inches; s = 4 ft.
• r = 8 yrds; s = 18 yrds.
• r = 10 inches; s = 40 inches
21. Exercises
Find the length of the arc on a circle of
radius r intercepted by the central
angle θ.
• r = 12 inches; θ = 45°
• r = 8 feet; θ = 225°
• r = 9 yards; θ = 315°
22. Exercises
Solve the following Word Problems.
1. A wheel is rotating at 250 revolutions
per minute. Find the angular speed in
radians per second.
2. A car with a tire diameter of 32 inches
is rotating at 450 revolutions per
minute. Find the speed of the car to the
nearest kilometer per hour.
23. 4. The radius of a wheel is 80 cm. If the
wheel rotates through an angle of
60°, how many centimeters does it
move? Express your answer to two
decimal places.
Exercises
24. 5. Suppose the figure
shows a highway sign
that warns of a railway
crossing. The lines that
form the cross pass
through the circles’
center and intersect at
right angles.
Exercises
25. If the radius of the circle is 24 inches,
Find the length of each of the four
arcs formed by the cross. Express
your final answer in the nearest
hundredths.
Exercises
26. • Assuming Earth to be a sphere of
radius 4000 miles, how many miles
north of the equator is Miami,
Florida, if it is 26 degrees north of
the equator? Show your complete
and organized computation.
Assignment
