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Angular Mechanics - Kinematics Contents:
Radians, Angles and Circles Linear and angular Qtys Conversions | Whiteboard Tangential Relationships Example | Whiteboard Angular Kinematics © Microsoft Encarta
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Angular Mechanics - Radians
Full circle: 360o = 2 Radians = s/r Radians = m/m = ? r s TOC
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Angular Mechanics - Angular Quantities Linear: Angular:
(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
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Conversions Radians Revolutions Rad/s Rev/min (RPM) = rev(2)
= (rev/min)(2 rad/rev)(min/60s) = (rev/s)(2 rad/rev) = (rad/s)(60 s/min)(rev/2 rad) TOC
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Whiteboards: Conversions 1 | 2 | 3 | 4 TOC
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How many radians in 3.16 revolutions?
rad = rev(2) rad = (3.16 rev)(2) = 19.9 rad W 19.9 rad
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If a drill goes through 174 radians, how many revolutions does it go through?
rev = rad/(2) rev = (174 rad)/(2) = 27.7 rev W 27.7 rev
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Convert 33 RPM to rad/s rad/s = (rev/min)(2 rad/rev)(min/60s)
W 3.5 rad/s
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rad/s = (rev/s)(2 rad/rev) rad/s = (12 rev/s)(2 rad/rev)
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 W 75 rad/s
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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* *Not in data packet TOC
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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) = s-1 r = .011m/2 = m v = r = ( s-1)(.0055 m) v = 5.8 m/s (show ‘em!) (pitching machines) TOC
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Tangential relationships
Whiteboards: Tangential relationships 1 | 2 | 3 | 4 | 5 | 6 TOC
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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 W 8.8 m/s
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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 W 95 rad/s
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A 0. 450 m radius marking wheel rolls a distance of 123. 2 m
A m radius marking wheel rolls a distance of m. What angle does the wheel rotate through? s = r 123.2 m = (.450 m) = (123.2 m)/(.450 m) = 274 rad W 274 rad
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(a) What is the linear acceleration?
A car with 0.36 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 W 3.0 m/s/s
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(b) What is the tire’s angular acceleration?
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) = Rad/s/s = 8.3 Rad/s/s W 8.3 Rad/s/s
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(c) What angle do the tires go through?
A car with 0.36 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 ( ) (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 = m s = r, m = (.36 m) = (121.5 m)/(.36 m) = Rad = 340 Rad W 340 Rad
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Linear: u + at = v ut + 1/2at2 = s u2 + 2as = v2 (u + v)t/2 = s
Angular Kinematics Linear: u + at = v ut + 1/2at2 = s u2 + 2as = v2 (u + v)t/2 = s Angular: f = i + t = it + 1/2t2 f 2 = i2 + 2 = (i + f)t/2* *Not in data packet TOC
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Example: My gyro spinner speeds up to 10,000 RPM, in 0.78 sec.
What is its angular acceleration? What angle does it go through? What distance does a point on the edge travel if the diameter is 1.1 cm? = ?, o= 0, t = .78 s = (10,000rev/min)(2 rad/rev)(1 min/60 sec) = s-1 = o + t s-1 = 0 + (.78s) = ( s-1)/(.78s) =1342.6=1300 rad/s/s (u + v)t/2 = s ( = (o + )t/2) ( s-1)(.78s)/2 = = 410 rad s = 2.25 m TOC
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Whiteboards: Angular Kinematics 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 TOC
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= (89 rad/s - 34 rad/s)/(2.5 sec) = 22 s-2
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 W 22 rad/s/s
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= (89 rad/s - 34 rad/s)/(2.5 sec) = 22 s-2
A drill slows from 145 rad/s to 54.0 rad/s with an angular acceleration of rad/s/s. Through what angle did it go? How many rotations? = o + t 89 rad/s = 34 rad/s + (2.5 sec) = (89 rad/s - 34 rad/s)/(2.5 sec) = 22 s-2 W 5030 radians, 801 rotations
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A motor going 45. 0 rad/s has an angular acceleration of 12
A motor going 45.0 rad/s has an angular acceleration of 12.4 rad/s/s for 3.7 seconds. What angle does it go through? = ot + 1/2t2 = (45.0s-1)(3.7s) + 1/2 (12.4s-2)(3.7s)2 = = 250 rad W 250 rad
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= (02 - (120 rad/s)2)/(2(18.85 rad)) = -381.97 = -380 rad/s/s
A wheel stops from 120 rad/s in 3.0 revolutions. (a) What is the angular acceleration? = (3.0)(2) = rad 2 = o2 + 2 = (2 - o2)/(2) = (02 - (120 rad/s)2)/(2(18.85 rad)) = = -380 rad/s/s W -380 rad/s/s
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A hard drive speeds up from rest to 4200. RPM in 3. 50 seconds
A hard drive speeds up from rest to RPM in 3.50 seconds. How many rotations does it make doing this? = 4200*2π/60 = rad/s θ = rad = rotations W 122.5 rotations
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A hard drive slows from 7200. RPM to rest in 16. 2 seconds
A hard drive slows from RPM to rest in 16.2 seconds. What distance does a point 3.10 cm from the center travel as it is slowing down? r = m wo = rad/s use no a θ = rad W 189 m
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A car with 0. 68 m diameter tires has an acceleration of 3. 60 m/s/s
A car with 0.68 m diameter tires has an acceleration of 3.60 m/s/s. Through what angle do the tires go when the car speeds up from 12.0 m/s to 32.0 m/s? How many rotations? r = m = (3.60 m/s/s)/(0.340 m) = rad/s/s w = v/r: rad/s to rad/s use no t W 359 radians, 57.2 rotations
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A drill speeds up from 16. 0 rot/sec to 72. 0 rot/sec in 10. 0 sec
A drill speeds up from 16.0 rot/sec to 72.0 rot/sec in 10.0 sec. How many rotations does it go through? use no a, you can leave it in rotations W 440 rotations
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