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Unformatted text preview: 107 Chapter 4 1. (a) The magnitude of r is 2 2 2   (5.0 m) ( 3.0 m) (2.0 m) 6.2 m. r (b) A sketch is shown. The coordinate values are in meters. 2. (a) The position vector, according to Eq. 41, is ? = ( 5.0 m) i + (9.0 m)j r . (b) The magnitude is 2 2 2 2 2 2   + + ( 5.0 m) (9.0 m) (0 m) 10 m. r x y z (c) Many calculators have polar rectangular conversion capabilities that make this computation more efficient than what is shown below. Noting that the vector lies in the xy plane and using Eq. 36, we obtain: 1 9.0 m tan 61 or 119 5.0 m where the latter possibility (119° measured counterclockwise from the + x direction) is chosen since the signs of the components imply the vector is in the second quadrant (d) The sketch is shown to the right. The vector is 119° counterclockwise from the + x direction. (e) The displacement is r r r where r is given in part (a) and ˆ (3.0 m)i. r Therefore, ? (8.0 m)i (9.0 m)j r . (f) The magnitude of the displacement is 2 2   (8.0 m) ( 9.0 m) 12 m. r (g) The angle for the displacement, using Eq. 36, is 1 8.0 m tan = 42 or 138 9.0 m 119 (5, 9) (5, 9) CHAPTER 4 108 where we choose the former possibility ( 42°, or 42° measured clockwise from + x ) since the signs of the components imply the vector is in the fourth quadrant. A sketch of r is shown on the right. 3. The initial position vector r o satisfies r r r o , which results in o ? ? ? ? (3.0j 4.0k)m (2.0i 3.0j 6.0k)m ( 2.0 m) i (6.0 m) j ( 10 m) k r r r . 4. We choose a coordinate system with origin at the clock center and + x rightward (toward the “3:00” position) and + y upward (toward “12:00”). (a) In unitvector notation, we have 1 2 ? (12 cm)i and ( 12 cm)j. r r Thus, Eq. 42 gives 2 1 ? ( 12 cm)i ( 12 cm)j. r r r The magnitude is given by 2 2   ( 12 cm) ( 12 cm) 17 cm. r (b) Using Eq. 36, the angle is 1 12 cm tan 45 or 135 . 12 cm We choose 135 since the desired angle is in the third quadrant. In terms of the magnitudeangle notation, one may write 2 1 ? ( 12 cm)i ( 12 cm)j. r r r (c) In this case, we have 1 2 ? ? ( 12 cm)j and (12 cm)j, and (24 cm)j. r r r Thus,   24 cm. r (d) Using Eq. 36, the angle is given by 1 24 cm tan 90 ....
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This document was uploaded on 04/22/2011.
 Spring '11

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