This document discusses linear velocity, angular velocity, and their relationships. It provides equations for calculating linear velocity, rotational velocity, and angular velocity. Examples are worked through, such as calculating the velocities of points on a spinning disk and the speed of a bicyclist based on wheel rotation speed. Students are encouraged to review this material and get help if needed, as the topic has historically been confusing.

1. 6.1 Angle Measure
Day 2
Linear Velocity and Angular Velocity
Matthew 22:37-39 Jesus replied: ‘Love the Lord your God
with all your heart and with all your soul and with all your
mind.’ This is the first and greatest commandment. And the
second is like it: ‘Love your neighbor as yourself.’

2. Linear Velocity is the rate at which the distance
traveled is changing.

3. Linear Velocity is the rate at which the distance
traveled is changing. (Straight-Line Velocity)

4. Linear Velocity is the rate at which the distance
traveled is changing. (Straight-Line Velocity)
d
V=
t

5. Linear Velocity is the rate at which the distance
traveled is changing. (Straight-Line Velocity)
d
V= just like in General Science
t

6. Linear Velocity is the rate at which the distance
traveled is changing. (Straight-Line Velocity)
d
V= just like in General Science
t
but in the case of Circular Motion, our distance is
actually an arc length. (Rotational Velocity)

7. Linear Velocity is the rate at which the distance
traveled is changing. (Straight-Line Velocity)
d
V= just like in General Science
t
but in the case of Circular Motion, our distance is
actually an arc length. (Rotational Velocity)
s
V=
t

8. Angular Velocity is the rate at which the central angle
θ is changing.

9. Angular Velocity is the rate at which the central angle
θ is changing.
θ
ω= θ in radians
t

10. Angular Velocity is the rate at which the central angle
θ is changing.
θ
ω= θ in radians
t
ω is the variable used
for angular velocity

11. Have these equations memorized:
d
Linear Velocity: V=
t
s
Rotational Velocity: V=
t
Angular Velocity:
θ
ω= θ in radians
t

12. A disk with a 12-inch diameter spins at a rate of 45
revolutions per minute (rpm). Find the angular and
linear velocities of a point at the edge of the disk in
radians per second and inches per second, respectively.

13. A disk with a 12-inch diameter spins at a rate of 45
revolutions per minute (rpm). Find the angular and
linear velocities of a point at the edge of the disk in
radians per second and inches per second, respectively.
θ 45 rev 2π rad 1 min
ω= = g g
t 1 min 1 rev 60 sec
3π rad rad
= or 1.5π
2 sec sec

14. A disk with a 12-inch diameter spins at a rate of 45
revolutions per minute (rpm). Find the angular and
linear velocities of a point at the edge of the disk in
radians per second and inches per second, respectively.
θ 45 rev 2π rad 1 min
ω= = g g
t 1 min 1 rev 60 sec
3π rad rad
= or 1.5π
2 sec sec
s r θ 6 ⋅ 45 ⋅ 2π in.
V= = =
t t 60 sec
in
= 9π
s

15. On that 12-inch diameter disk spinning at 45 rpm,
compare the linear velocities of two points ... one 2
inches from the center and the other 5 inches from the
center.

16. On that 12-inch diameter disk spinning at 45 rpm,
compare the linear velocities of two points ... one 2
inches from the center and the other 5 inches from the
center.
s rθ
V= =
t t

17. On that 12-inch diameter disk spinning at 45 rpm,
compare the linear velocities of two points ... one 2
inches from the center and the other 5 inches from the
center.
s rθ
V= =
t t
2 ⋅ 45 ⋅ 2π
V2in =
60

18. On that 12-inch diameter disk spinning at 45 rpm,
compare the linear velocities of two points ... one 2
inches from the center and the other 5 inches from the
center.
s rθ
V= =
t t
2 ⋅ 45 ⋅ 2π
V2in =
60
in
V2in = 3π
s

19. On that 12-inch diameter disk spinning at 45 rpm,
compare the linear velocities of two points ... one 2
inches from the center and the other 5 inches from the
center.
s rθ
V= =
t t
2 ⋅ 45 ⋅ 2π 5 ⋅ 45 ⋅ 2π
V2in = V5in =
60 60
in
V2in = 3π
s

20. On that 12-inch diameter disk spinning at 45 rpm,
compare the linear velocities of two points ... one 2
inches from the center and the other 5 inches from the
center.
s rθ
V= =
t t
2 ⋅ 45 ⋅ 2π 5 ⋅ 45 ⋅ 2π
V2in = V5in =
60 60
in in
V2in = 3π V5in = 7.5π
s s

21. On that 12-inch diameter disk spinning at 45 rpm,
compare the linear velocities of two points ... one 2
inches from the center and the other 5 inches from the
center.
s rθ
V= =
t t
2 ⋅ 45 ⋅ 2π 5 ⋅ 45 ⋅ 2π
V2in = V5in =
60 60
in in
V2in = 3π V5in = 7.5π
s s
the point further from the center travels faster!

22. Same two points ... same disk ... same spin rate.
Compare the angular velocities.

23. Same two points ... same disk ... same spin rate.
Compare the angular velocities.
θ
ω=
t

24. Same two points ... same disk ... same spin rate.
Compare the angular velocities.
θ
ω=
t
45 rev 2π rad
ω 2in = ⋅
60 sec 1 rev

25. Same two points ... same disk ... same spin rate.
Compare the angular velocities.
θ
ω=
t
45 rev 2π rad
ω 2in = ⋅
60 sec 1 rev
rad
ω 2in = 1.5π
s

26. Same two points ... same disk ... same spin rate.
Compare the angular velocities.
θ
ω=
t
45 rev 2π rad 45 rev 2π rad
ω 2in = ⋅ ω 5in = ⋅
60 sec 1 rev 60 sec 1 rev
rad
ω 2in = 1.5π
s

27. Same two points ... same disk ... same spin rate.
Compare the angular velocities.
θ
ω=
t
45 rev 2π rad 45 rev 2π rad
ω 2in = ⋅ ω 5in = ⋅
60 sec 1 rev 60 sec 1 rev
rad rad
ω 2in = 1.5π ω 5in = 1.5π
s s

28. Same two points ... same disk ... same spin rate.
Compare the angular velocities.
θ
ω=
t
45 rev 2π rad 45 rev 2π rad
ω 2in = ⋅ ω 5in = ⋅
60 sec 1 rev 60 sec 1 rev
rad rad
ω 2in = 1.5π ω 5in = 1.5π
s s
same angular velocities

29. Mr. Kindschi is riding a bicycle whose wheels are 30
inches in diameter. If the wheels rotate at 150 rpm,
find the speed at which he is traveling in mph.

30. Mr. Kindschi is riding a bicycle whose wheels are 30
inches in diameter. If the wheels rotate at 150 rpm,
find the speed at which he is traveling in mph.
150 rev 2π ⋅15 in 60 min 1 ft 1 mile
⋅ ⋅ ⋅ ⋅
1 min 1 rev 1 hr 12 in 5280 ft

31. Mr. Kindschi is riding a bicycle whose wheels are 30
inches in diameter. If the wheels rotate at 150 rpm,
find the speed at which he is traveling in mph.
150 rev 2π ⋅15 in 60 min 1 ft 1 mile
⋅ ⋅ ⋅ ⋅
1 min 1 rev 1 hr 12 in 5280 ft
≈ 13.4 mph

32. This topic has often been confusing for Pre-Cal students
in the past. If you have trouble with this assignment,
COME IN TOMORROW at 7:15!! I will be available for
you in the CAD Lab area (my office area). Don’t simply
wait for class ... we are moving on!!
HW #2
A falling drop at last will carve a stone.
Lucretius