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chap03 - 19 Chapter 3 Velocity 3.1 The position vector of a...

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19 Chapter 3 Velocity 3.1 The position vector of a point is given by the equation 100 j t e π = R , where R is in inches. Find the velocity of the point at 0.40 s. t = ( 29 100 j t t e π = R ( 29 100 j t t j e π π = R & ( 29 ( 29 0.40 0.40 100 100 cos0.40 sin 0.40 100 sin 72 100 cos72 j j e j j j π π π π π π π = = + = - ° + ° R & ( 29 0.40 298.783 97.080 314.159 162 in/s j = - + = ° R & Ans. 3.2 The equation ( 29 2 10 4 -j t/ R t e π = + defines the path of a particle. If R is in meters, find the velocity of the particle at 20 s. t = ( 29 ( 29 2 /10 4 j t t t e π - = + R ( 29 ( 29 ( 29 /10 2 /10 2 /10 4 j t j t t te j t e π π π - - = - + R & ( 29 ( 29 ( 29 20/10 2 20/10 2 2 20 40 /10 20 4 40 40.4 j j j j e j e e j e π π π π π π - - - - = - + = - R & ( 29 20 40.000 126.920 133.074 72.5 m/s j = - = ∠ - ° R & Ans. 3.3 If automobile A is traveling south at 55 mi/h and automobile B north 60 ° east at 40 mi/h, what is the velocity difference between B and A ? What is the apparent velocity of B to the driver of A ? ˆ 55 90 55 mi/h A = ∠ - ° = - V j ˆ ˆ 40 30 34.641 20 mi/h B = ° = + V i j ˆ ˆ 34.641 75 mi/h BA B A = - = + V V V i j 82.613 65.2 =82.613 N 24.8 E mi/h BA = ° ° V Ans Naming B as car 3 and A as car 2, we have 2 B A = V V since 2 is translating. Then 3 3 2 / 2 B B B BA = - = V V V V 3 / 2 82.613 65.2 =82.613 N 24.8 E mi/h B = ° ° V Ans.
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20 3.4 In the figure, wheel 2 rotates at 600 rev/min and drives wheel 3 without slipping. Find the velocity difference between points B and A . ( 29 ( 29 2 600 rev/min 2 rad/rev 60 s/min 20 rad/s cw π ϖ π = = 2 2 2 80 251 in/s AO AO V R ϖ π = = = B A BA = + V V V 223 90 in/s B = ° V Ans. 3.5 Two points A and B , located along the radius of a wheel (see figure), have speeds of 80 and 140 m/s, respectively. The distance between the points is 300 mm BA R = . ( a ) What is the diameter of the wheel? (b) Find , , AB BA V V and the angular velocity of the wheel. ( 29 ( 29 ˆ ˆ ˆ 140 80 60 m/s BA B A = - = - - - = - V V V j j j Ans.
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21 ( 29 ( 29 ˆ ˆ ˆ 80 140 60 m/s AB A B = - = - - - = V V V j j j Ans. 2 60 m/s 200 rad/s cw 0.300 m BA BA V R ϖ = = = Ans. 2 2 2 140 m/s 0.700 m 200 rad/s BO BO V R ϖ = = = ( 29 2 Dia 2 2 0.7 m 1.4 m 1 400 mm BO R = = = = Ans.
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22 3.6 A plane leaves point B and flies east at 350 mi/h. Simultaneously, at point A , 200 miles southeast (see figure), a plane leaves and flies northeast at 390 mi/h. (a) How close will the planes come to each other if they fly at the same altitude? (b) If they both leave at 6:00 p.m., at what time will this occur? ˆ ˆ 390 45 mi/h 276 276 A = ° = + V i j ; ˆ 350 B = V i ˆ ˆ 74 276 BA B A = - = - V V V i j At initial time ( 29 ˆ ˆ 0 200 120 mi 100 173 BA = ° = - + R i j At later time ( 29 ( 29 ( 29 ˆ ˆ ( ) 0 100 74 173 276 BA BA BA t t t t = + = - + + - R R V i j To find the minimum of this: ( 29 ( 29 2 2 2 100 74 173 276 BA R t t = - + + - ( 29 ( 29 ( 29 ( 29 2 2 100 74 74 2 173 276 276 0 BA dR dt t t = - + + - - = 163 120 110 376 0 t - = 0.677 h 41 min t = = or 6:41p.m. Ans. ( 29 ˆ ˆ 0.677 49.8 13.4 51.5 165 mi BA = - - = ∠- ° R i j Ans. 3.7 To the data of Problem 3.6, add a wind of 30 mi/h from the west. (a) If A flies the same heading, what is its new path? (b) What change does the wind make in the results of Problem 3.6? With the added wind ˆ ˆ 306 276 412 42 mi/h A = + = ° V i j Since the velocity is constant, the new path is a straight line at N 48º E. Ans.
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