Angular Mechanics - Kinematics
Contents:
•Radians, Angles and Circles
•Linear and angular Qtys
•Conversions | Whiteboard
•Tangential Relationships
• Example | Whiteboard
•Angular Kinematics
• Example | Whiteboard
© Microsoft Encarta
Angular Mechanics - Radians
r

s
Full circle:
360o = 2 Radians
 = s/r
Radians = m/m = ?
TOC
Angular Mechanics - Angular Quantities
Linear: Angular:
(m) s  - Angle (Radians)
(m/s) u i - Initial angular velocity (Rad/s)
(m/s) v  f - Final angular velocity (Rad/s)
(m/s/s) a  - Angular acceleration (Rad/s/s)
(s) t t
- Uh, time (s)
TOC
Conversions
Radians
Revolutions
Rad/s
Rad/s
Rev/min
(RPM)
= rev(2)
= rad/(2)
= (rev/min)(2 rad/rev)(min/60s)
= (rev/s)(2 rad/rev)
= (rad/s)(60 s/min)(rev/2 rad)
TOC
Whiteboards:
Conversions
1|2|3|4
TOC
How many radians in 3.16
revolutions?
rad = rev(2)
rad = (3.16 rev)(2) = 19.9 rad
19.9 rad
W
If a drill goes through 174 radians,
how many revolutions does it go
through?
rev = rad/(2)
rev = (174 rad)/(2) = 27.7 rev
27.7 rev
W
Convert 33 RPM to rad/s
rad/s = (rev/min)(2 rad/rev)(min/60s)
= (33rev/min)(2 rad/rev)(min/60s)
rad/s = 3.5 rad/s
3.5 rad/s
W
Convert 12 rev/s to rad/s
rad/s = (rev/s)(2 rad/rev)
rad/s = (12 rev/s)(2 rad/rev)
rad/s = 75 rad/s
75 rad/s
W
Angular Mechanics - Tangential Relationships
Linear: Tangential: (at the edge of the wheel)
(m) s = r
- Displacement
(m/s) v = r
- Velocity
(m/s/s) a = r
- Acceleration*
*Not in data packet
TOC
Example: s = r, v = r, a = r
A certain gyro spinner has an angular
velocity of 10,000 RPM, and a
diameter of 1.1 cm. What is the
tangential velocity at its edge?
 = (10,000rev/min)(2 rad/rev)(1 min/60 sec)
 = 1047.19 s-1
r = .011m/2 = .0055 m
v = r = (1047.19 s-1)(.0055 m)
v = 5.8 m/s
(show ‘em!)
(pitching machines)
TOC
Whiteboards:
Tangential relationships
1|2|3|4|5|6
TOC
What is the tangential velocity of a
13 cm diameter grinding wheel
spinning at 135 rad/s?
v = r, r = .13/2 = .065 m
v = (135 rad/s)(.065 m) = 8.8 m/s
8.8 m/s
W
What is the angular velocity of a
57 cm diameter car tire rolling at
27 m/s?
v = r, r = .57/2 = .285 m
27 m/s = (.285 m)
 = (27 m/s)/ (.285 m) = 95 rad/s
95 rad/s
W
A .450 m radius marking wheel
rolls a distance of 123.2 m. What
angle does the wheel rotate
through?
s = r
123.2 m = (.450 m)
 = (123.2 m)/(.450 m) = 274 rad
274 rad
W
A car with .36 m radius tires
speeds up from 0 to 27 m/s in 9.0
seconds.
(a) What is the linear acceleration?
v = u + at
27 m/s = 0 + a(9.0s)
a = (27 m/s)/(9.0s) = 3.0 m/s/s
3.0 m/s/s
W
A car with .36 m radius tires speeds up
from 0 to 27 m/s in 9.0 seconds.
(a) a = 3.0 m/s/s
(b) What is the tire’s angular
acceleration?
a = r
(3.0 m/s/s) = (.36 m)
 = (3.0 m/s/s)/(.36 m) = 8.3333 Rad/s/s
 = 8.3 Rad/s/s
8.3 Rad/s/s
W
A car with .36 m radius tires speeds up
from 0 to 27 m/s in 9.0 seconds.
(a) a = 3.0 m/s/s
(b)  = 8.3 Rad/s/s (8.33333333)
(c) What angle do the tires go through?
s = r, s = (u + v)t/2, r = .36 m
s = (27 m/s + 0)(9.0 s)/2 = 121.5 m
s = r, 121.5 m = (.36 m)
 = (121.5 m)/(.36 m) = 337.5 Rad
 = 340 Rad
340 Rad
W
Angular Mechanics - Angular kinematics
Linear:
s/t = v
v/t = a
u + at = v
1
2
ut + /2at = s
u2 + 2as = v2
(u + v)t/2 = s
Angular:
 = /t
 = /t*
 = o + t
1
2
 = ot + /2t
2 = o2 + 2
 = (o + )t/2*
*Not in data packet
TOC
Example: My gyro spinner speeds up to
10,000 RPM, in .78 sec. What is its angular
accel., and what angle does it go through?
= ?, o= 0, t = .78 s
 = (10,000rev/min)(2 rad/rev)(1 min/60 sec)
 = 1047.19 s-1
 = o + t
1047.19 s-1 = 0 + (.78s)
 = (1047.19 s-1)/(.78s) =1342.6=1300 rad/s/s
(u + v)t/2 = s
( = (o + )t/2)
(0 + 1047.19 s-1)(.78s)/2 = 408.4 = 410 rad
TOC
Whiteboards:
Angular Kinematics
1|2|3|4|5|6|7
TOC
Use the formula  = /t to convert
the angular velocity 78 RPM to rad/s.
Hint: t = 60 sec,  = 78(2)
 = /t
 = (78(2))/(60 sec) = 8.2 rad/s
8.2 rad/s
W
A turbine speeds up from 34 rad/s to
89 rad/s in 2.5 seconds. What is the
angular acceleration?
 = o + t
89 rad/s = 34 rad/s + (2.5 sec)
 = (89 rad/s - 34 rad/s)/(2.5 sec) = 22 s-2
22 rad/s/s
W
A turbine speeds up from 34 rad/s to
89 rad/s in 2.5 seconds. What is the
angular acceleration? (b) What angle
does it go through?
(u + v)t/2 = s
(34 rad/s + 89 rad/s)(2.5 s)/2 = 150 rad
150 rad
W
A wheel stops from 120 rad/s in 3.0
revolutions. (a) What is the angular
acceleration?
 = (3.0)(2) = 18.85 rad
2 = o2 + 2
 = (2 - o2)/(2)
 = (02 - (120 rad/s)2)/(2(18.85 rad))
 = -381.97 = -380 rad/s/s
-380 rad/s/s
W
A wheel stops from 120 rad/s in 3.0
revolutions. (a) What is the angular
acceleration? (b) What time did it take?
 = 381.97 = -380 rad/s/s
v/t = a,
t = v/a = (120 rad/s)/t
= (120 s-1)/(381.97 s-2) = .31 sec
.31 s
W
A motor going 45.0 rad/s has an angular
acceleration of 12.4 rad/s/s for 3.7
seconds. (a) What is the final velocity?
 = o + t
 = 45.0 rad/s + (12.4 rad/s/s)(3.7 s) =
 = 90.88 = 91 rad/s
91 rad/s
W
A motor going 45.0 rad/s has an angular
acceleration of 12.4 rad/s/s for 3.7
seconds. (a) What is the final velocity?
(b) What angle does it go through?
 = ot + 1/2t2
 = (45.0s-1)(3.7s) + 1/2 (12.4s-2)(3.7s)2
 = 251.378 = 250 rad
250 rad
W