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Unformatted text preview: Chapter 3 1. The x and the y components of a vector a lying on the xy plane are given by cos , sin x y a a a a = = where | | a a = is the magnitude and is the angle between a and the positive x axis. (a) The x component of a is given by a x = 7.3 cos 250 = 2.5 m. (b) and the y component is given by a y = 7.3 sin 250 = 6.9 m. In considering the variety of ways to compute these, we note that the vector is 70 below the x axis, so the components could also have been found from a x = 7.3 cos 70 and a y = 7.3 sin 70. In a similar vein, we note that the vector is 20 to the left from the y axis, so one could use a x = 7.3 sin 20 and a y = 7.3 cos 20 to achieve the same results. 2. The angle described by a full circle is 360 = 2 rad, which is the basis of our conversion factor. (a) ( 29 2 rad 20.0 20.0 0.349 rad 360 = = . (b) ( 29 2 rad 50.0 50.0 0.873 rad 360 = = (c) ( 29 2 rad 100 100 1.75 rad 360 = = (d) ( 29 360 0.330 rad = 0.330 rad 18.9 2 rad = (e) ( 29 360 2.10 rad = 2.10 rad 120 2 rad = (f) ( 29 360 7.70 rad = 7.70 rad 441 2 rad = 79 CHAPTER 3 3. A vector a can be represented in the magnitude-angle notation ( a , ), where 2 2 x y a a a = + is the magnitude and 1 tan y x a a - = is the angle a makes with the positive x axis. (a) Given A x = - 25.0 m and A y = 40.0 m, 2 2 ( 25.0 m) (40.0 m) 47.2 m A =- + = (b) Recalling that tan = tan ( + 180), tan 1 [40/ ( 25)] = 58 or 122. Noting that the vector is in the third quadrant (by the signs of its x and y components) we see that 122 is the correct answer. The graphical calculator shortcuts mentioned above are designed to correctly choose the right possibility. 4. (a) With r = 15 m and = 30, the x component of r is given by r x = r cos = 15 cos 30 = 13 m. (b) Similarly, the y component is given by r y = r sin = 15 sin 30 = 7.5 m. 5. The vector sum of the displacements d storm and . d new must give the same result as its originally intended displacement o . =120j d where east is i , north is j, and the assumed length unit is km. Thus, we write storm new 100i, i + j. d d A B = = r r (a) The equation storm new o d d d + = readily yields A = 100 km and B = 120 km. The magnitude of d new is therefore 2 2 156km A B + = . (b) And its direction is tan 1 ( B / A ) = 50.2 or 180 + ( 50.2) = 129.8. We choose the latter value since it indicates a vector pointing in the second quadrant, which is what we expect here. The answer can be phrased several equivalent ways: 129.8 counterclockwise from east, or 39.8 west from north, or 50.2 north from west....
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- Fall '08