Angular Mechanics - Kinematics

Contents:•Radians, Angles and Circles•Linear and angular Qtys•Conversions | Whiteboard•Tangential Relationships

• Example | Whiteboard•Angular Kinematics

• Example | Whiteboard

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Angular Mechanics - Radians

r

sFull circle:

360o = 2 Radians = s/rRadians = m/m = ?

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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)

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Conversions

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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)

Whiteboards: Conversions1 | 2 | 3 | 4

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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:(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)

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Whiteboards: Tangential relationships

1 | 2 | 3 | 4 | 5 | 6

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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 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

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Whiteboards: Angular Kinematics1 | 2 | 3 | 4 | 5 | 6 | 7

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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

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 + 1/2t2

= (45.0s-1)(3.7s) + 1/2 (12.4s-2)(3.7s)2

= 251.378 = 250 rad

250 rad W