the platform, we expect to find < 41°. SOLUTIONPutting the above relationships together, we have
Chapter 1 Problems 132111coscos41 or cos41coscoscos410.75 mcoscos41coscos41328.0 mdhhLLdLLdLdL23. SSMREASONING a. Since the two force vectors Aand Bhave directions due west and due north, they are perpendicular. Therefore, the resultant vector F = A + Bhas a magnitude given by the Pythagorean theorem: F2= A2+ B2. Knowing the magnitudes of Aand B, we can calculate the magnitude of F. The direction of the resultant can be obtained using trigonometry. b. For the vector F= A –Bwe note that the subtraction can be regarded as an addition in the following sense: F= A + (–B). The vector –Bpoints due south, opposite the vector B, so the two vectors are once again perpendicular and the magnitude of Fagain is given by the Pythagorean theorem. The direction again can be obtained using trigonometry. SOLUTIONa. The drawing shows the two vectors and the resultant vector. According to the Pythagorean theorem, we have FABFABF2222222445 N325 N551 NbgbgUsing trigonometry, we can see that the direction of the resultant is tantan.FHGIKJBAor =325 Nnorth of west1445 N36 1b. Referring to the drawing and following the same procedure as in part a, we find FHGIKJFABFABBA22222221445 N325551 N325445 N36 1bgbgbgbgor Nor =Nsouth of westtantan._____________________________________________________________________________________________B North A (a) (b) F North A –B F
14INTRODUCTION AND MATHEMATICAL CONCEPTS 24. REASONINGSince the initial force and the resultant force point along the east/west line, the second force must also point along the east/west line. The direction of the second force is not specified; it could point either due east or due west, so there are two answers. We use “N” to denote the units of the forces, which are specified in newtons.SOLUTIONIf the second force points due east ,both forces point in the same direction and the magnitude of the resultant force is the sum of the two magnitudes: F1+ F2= FR. Therefore, F2= FR–F1= 400 N –200 N = 200 N If the second force points due west , the two forces point in opposite directions, and the magnitude of the resultant force is the difference of the two magnitudes: F2–F1= FR. Therefore, F2= FR+ F1= 400 N + 200 N =600 N _____________________________________________________________________________________________25. SSMREASONINGFor convenience, we can assign due east to be the positive direction and due west to be the negative direction. Since all the vectors point along the same east-west line, the vectors can be added just like the usual algebraic addition of positive and negative scalars. We will carry out the addition for all of the possible choices for the two vectors and identify the resultants with the smallest and largest magnitudes.
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