chap02 - 7 Chapter 2 Position and Displacement 2.1 Describe...

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7 Chapter 2 Position and Displacement 2.1 Describe and sketch the locus of a point A which moves according to the equations ( 29 cos 2 x A R at t = π , ( 29 sin 2 y A R at t π = , 0 z A R = . 2.2 Find the position difference from point P to point Q on the curve 2 16 y x x = + - , where 2 x P R = and 4 x Q R = . ( 29 2 2 2 16 10 y P R = + - = - ; ˆ ˆ 2 10 P = - R i j ( 29 2 4 4 16 4 y Q R = + - = ; ˆ ˆ 4 4 Q = + R i j ˆ ˆ 2 14 14.142 81.9 QP Q P = - = + = ° R R R i j Ans. 2.3 The path of a moving point is defined by the equation 2 2 28 y x = - . Find the position difference from point P to point Q if 4 x P R = and 3 x Q R = - .
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8 ( 29 2 2 4 28 4 y P R = - = ; ˆ ˆ 4 4 P = + R i j ( 29 2 2 3 28 10 y Q R = - - = - ; ˆ ˆ 3 10 Q = - - R i j ˆ ˆ 7 14 15.652 243.4 QP Q P = - = - - = ° R R R i j Ans. 2.4 The path of a moving point P is defined by the equation 3 60 /3 y x = - . What is the displacement of the point if its motion begins when 0 x P R = and ends when 3 x P R = ? ( 29 ( 29 3 0 60 0 /3 60 y P R = - = ; ( 29 ˆ 0 60 P = R j ( 29 ( 29 3 3 60 3 /3 51 y P R = - = ; ( 29 ˆ ˆ 3 3 51 P = + R i j ( 29 ( 29 ˆ ˆ 3 0 3 9 9.487 71.6 P P P = - = - = ∠ - ° ΔR R R i j Ans. 2.5 If point A moves on the locus of Problem 2.1, find its displacement from t = 2 to t =2.5. ( 29 ˆ ˆ ˆ 2.0 2.0 cos4 2.0 sin 4 2.0 A a a a π π = + = R i j i ( 29 ˆ ˆ ˆ 2.5 2.5 cos5 2.5 sin5 2.5 A a a a π π = + = - R i j i ( 29 ( 29 ˆ 2.5 2.0 4.5 A A A a = - = - ΔR R R i Ans. 2.6 The position of a point is given by the equation 2 100 j t e π = R . What is the path of the point? Determine the displacement of the point from t = 0.10 to t = 0.40. The point moves in a circle of radius 100 with its center at the origin. Ans. ( 29 0.628 ˆ ˆ 0.10 100 80.902 58.779 j e = = + R i j ( 29 2.513 ˆ ˆ 0.40 100 80.902 58.779 j e = = - + R i j ( 29 ( 29 ˆ 0.40 0.10 161.803 161.803 180 = - = - = ° ΔR R R i Ans.
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9 2.7 The equation ( 29 2 /10 4 j t t e - π = + R defines the position of a point. In which direction is the position vector rotating? Where is the point located when t = 0? What is the next value t can have if the direction of the position vector is to be the same as it is when t = 0? What is the displacement from the first position of the point to the second? Since the polar angle for the position vector is /10 j t θ = - π , then / d dt θ is negative and therefore the position vector is rotating clockwise. Ans. ( 29 ( 29 2 0 0 0 4 4 0 j e - = + = ∠ ° R Ans. The position vector will next have the same direction when /10 2 t π π = , that is, when t =20. Ans. ( 29 ( 29 2 2 20 20 4 404 0 j e π - = + = ∠ ° R ( 29 ( 29 20 0 400 0 = - = ∠ ° R R R Ans. 2.8 The location of a point is defined by the equation ( 29 2 / 30 4 2 j t t e π = + R , where t is time in seconds. Motion of the point is initiated when t = 0. What is the displacement during the first 3 s? Find the change in angular orientation of the position vector during the same time interval. ( 29 ( 29 0 ˆ 0 0 2 2 0 2 j e = + = ∠ ° = R i ( 29 ( 29 9/ 30 ˆ ˆ 3 12 2 14 54 8.229 11.326 j e π = + = ° = + R i j ( 29 ( 29 ˆ ˆ 3 0 6.229 11.326 12.926 61.2 = - = + = ° ΔR R R i j Ans. 54 0 54 ccw θ = °- ° = ° Ans.
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10 2.9 Link 2 in the figure rotates according to the equation / 4 t θ = π . Block 3 slides outward on link 2 according to the equation 2 2 r t = + . What is the absolute displacement 3 P R from t = 1 to t = 2 ? What is the apparent displacement 3/ 2 P R ?
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