auto generate barcode vb net Linear and Angular Velocity in .NET framework

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1.7 Linear and Angular Velocity
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Consider an object traveling at a constant velocity along a circular arc of radius r. Let s be the length of the arc traveled in time t. Let 2 be the angle (in radian measure) corresponding to arc length s. Linear velocity measures how fast the object travels. The linear velocity, v, of an object is computed by arc length s n t. time Angular velocity measures how fast the angle changes. The angular velocity, (the lower-case Greek central angle in radians u letter omega) of the object, is computed by v time t. The relationship between the linear velocity v and the angular velocity for an object with radius r is v where (NOTE: rv
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is measured in radians per unit of time and v is distance per unit of time. v and use the same unit of time and r and v use the same linear unit.)
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CHAPTER 1 Angles and Applications
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EXAMPLE 1.7 A bicycle with 20-in wheels is traveling down a road at 15 mi/h. Find the angular velocity of the wheel in revolutions per minute.
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Because the radius is 10 in and the angular velocity is to be in revolutions per minute (r/min), change the linear velocity 15 mi/h to units of in/min. v v To change 15 15 mi # 5280 ft # 12 in # 1 h mi h 1 h 1 mi 1 ft 60 min 15,840 rad rad v 1584 r 10 min min 15,840 in min
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to r/min, we multiply by 1/2 revolution per radian (r/rad). v 1584 rad min 1584 rad # 1 r 1 min 2p rad 792 r p min or 252 r/min
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EXAMPLE 1.8 A wheel that is drawn by a belt is making 1 revolution per second (r/s). If the wheel is 18 cm in diameter, what is the linear velocity of the belt in cm/s
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r 1s v
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1 # 2p rad 1 1 r rv 9(2p)
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2p rad/s 18p cm/s or 57 cm/s (See Probs. 1.11 to 1.15.)
SOLVED PROBLEMS
Use the directions for rounding stated on page 2. 1.1 Express each of the following angles in radian measure:
(a) 30 , (b) 135 , (c) 25 30 , (d) 42 24 35 , (e) 165.7 , (f) 3.85 , (g) 205 , (h) 18 30 , (i) 0.21 (a) 30 (b) 135 (c) 25 30 (d) 42 24 35 (e) 165.7 (f) (g) (h) (i) 3.85 205 18 30 0.21 30( /180) rad 135( /180) rad 25.5 42.41 /6 rad or 0.5236 rad 3 /4 rad or 2.3562 rad 0.4451 rad 0.7402 rad
25.5( /180) rad
42.41( /180) rad 2.8920 rad
165.7( /180) rad 3.85( /180) rad ( 205)( /180) rad 18.01 ( 0.21)( /180) rad
0.0672 rad 3.5779 rad 0.3143 rad 0.0037 rad
( 18.01)( /180) rad
1.2 Express each of the following angles in degree measure:
(a) /3 rad, (b) 5 /9 rad, (c) 2/5 rad, (d) 4/3 rad, (e) (f) 2 rad, (g) 1.53 rad, (h) 3 /20 rad, (i) 7 rad (a) /3 rad ( /3)(180 / ) (5 /9)(180 / ) (2/5)(180 / ) (4/3)(180 / ) (2)(180 / ) 60 100 22.92 or 22 55.2 or 22 55 12 76.39 or 76 23.4 or 76 23 24 22.5 or 22 30 87.66 or 87 39.6 or 87 39 36 27 1260 72 / 240 / (b) 5 /9 rad (c) 2/5 rad (d) 4/3 rad (e) /8 rad (f) 2 rad (h) (i) /8 rad,
( /8)(180 / ) (1.53)(180 / ) ( 7 )(180 / )
114.59 or 114 35.4 or 114 35 24
(g) 1.53 rad 7 rad
3 /20 rad
( 3 /20)(180 / )
CHAPTER 1 Angles and Applications
1.3 The minute hand of a clock is 12 cm long. How far does the tip of the hand move during 20 min
During 20 min the hand moves through an angle distance s r 12(2 /3) 8 cm 25.1 cm. 120 2 /3 rad and the tip of the hand moves over a
1.4 A central angle of a circle of radius 30 cm intercepts an arc of 6 cm. Express the central angle radians and in degrees.
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