angular mechanics - kinematics contents: radians, angles and circles linear and angular qtys...
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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 - Angular Quantities
Linear:(m) s
(m/s) u(m/s) v
(m/s/s) a(s) t
Angular: - Angle (Radians)
i - Initial angular velocity (Rad/s)
f - Final angular velocity (Rad/s)
- Angular acceleration (Rad/s/s)
t - Uh, time (s)
TOC
Conversions
TOC
RadiansRevolutions
Rad/sRad/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)
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:(m) s
(m/s) v(m/s/s) a
Tangential: (at the edge of the wheel)
= r - Displacement
= r - Velocity
= r - Acceleration*
TOC*Not in data packet
Example: s = r, v = r, a = rA 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 mv = r = (1047.19 s-1)(.0055 m)v = 5.8 m/s (show ‘em!)(pitching machines)
TOC
What is the tangential velocity of a 13 cm diameter grinding wheel spinning at 135 rad/s?
v = r, r = .13/2 = .065 mv = (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 m27 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 = r123.2 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 + at27 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 ms = (27 m/s + 0)(9.0 s)/2 = 121.5 ms = 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 = vv/t = au + at = v
ut + 1/2at2 = su2 + 2as = v2
(u + v)t/2 = s
Angular: = /t = /t* = o + t = ot + 1/2t2
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
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 + t89 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 rad2 = o
2 + 2 = (2 - o
2)/(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