49 34 30 cos ⬔CBA1 cos ⬔CBA 2 120° cos 1 10.5 2⬔CBA The angle that is required is ⬔DAB, the supplement of ⬔CBA. ⬔DAB 60°, and the angle between the 3 N and 5 N force is 60°. IN SUMMARY Key Ideas • Problems involving forces can be solved using strategies involving vectors. • When two or more forces are applied to an object, the net effect of the forces can be represented by the resultant vector determined by adding the vectors that represent each of the forces. • A system is in a state of equilibrium when the net effect of all the forces acting on an object causes no movement of the object. Need to Know ! ! ! ! ! • F F1 F2 is the resultant of F1 and F2 . ! ! ! ! ! • F 1F1 F2 2 is the equilibrant of F1 and F2 . ! ! ! ! ! ! ! • If a b c 0 , then a , b , and c are in a state of equilibrium. Exercise 7.1 PART A 1. a. Name some common household items that have approximate weights of 10 N, 50 N, and 100 N. b. What is your weight in newtons? 2. Three forces of 10 N, 20 N, and 30 N are in a state of equilibrium. a. Draw a sketch of these three forces. b. What is the angle between the equilibrant and each of the smaller forces? 3. Two forces of 10 N and 20 N are acting on an object. How should these forces be arranged to produce the largest possible resultant? 4. Explain in your own words why three forces must lie in the same plane if they are acting on an object in equilibrium. 362 7 . 1 V E C TO R S A S F O R C E S NEL (Assume that the force exerted by your arms is the same in both cases. 2 N. CHAPTER 7 363 . Three forces. Draw a sketch of these three forces. such that one component force makes an angle of 15° with the 10 N force. Resolve a force of 10 N into two forces perpendicular to each other. 10 N. are in a state of equilibrium. V5 N. 41 N c.8 N. Determine the resultant and equilibrant of each pair of forces acting on an object. 7 N. f 1 has a magnitude of 5 N acting due east. 14 N NEL 12. ! ! a. 3 N. a. f 1 has a magnitude of 9 N acting due west. 6. A second force. 9 N. b. f 1 .) 10 kg 30° ground 9N 5N 10 N 11. 9 N. 9 N d. and f 2 has a magnitude of 12 N acting due north. Determine the angle between the two smallest forces. A 10 kg block lies on a smooth ramp that is inclined at 30°. A 9. with magnitudes 13 N. and f 2 has a magnitude of 12 N acting due south. Which of the following sets of forces acting on an object could produce equilibrium? a. parallel to the ramp. of 2 ! f magnitude 8 N acts at 60° to .K PART B 5. of magnitude 6 N acts on particle P. 19 N 7. explain why it is easier to do chin-ups when your hands are 30 cm apart instead of 90 cm apart. 10. . 9 N. What force. and 8 N. Determine the resultant and equilibrant 1 ! ! of f 1 and f 2 .) ! ! f 8. would prevent the block from moving? (Assume that 1 kg exerts a force of 9. Four forces of magnitude 5 N. Using a vector diagram. 4 N b. 40 N. 6 N. and 14 N are arranged as shown in the diagram at the left. A force. Determine the resultant of these forces. 10 N. ! ! b. The smaller tug is 15° off the port bow. Determine the angle that the equilibrant makes with each of the three forces. 8 N. 1 V E C TO R S A S F O R C E S NEL . and f 4 . as shown. with @ f1 @ 30 N and ! @ f2 @ 40 N. Three forces of 5 N. and @ f1 @ 24 N. f 2 . act at right angles to each other. as shown.! ! 13. A mass of 20 kg is suspended from a ceiling by two lengths of rope that make angles of 30° and 45° with the ceiling. T 35 N 40 N f2 O 25 N 45° 30 N f3 f1 b. Determine @ f2 @ . Determine the tension in each of the ropes. Prove that @ f1 f2 @ @ f1 @ 2 @ f2 @ 2 2 @ f1 @ @ f2 @ cos u. Two forces. and the angle between 1 ! f 1 and the equilibrant. f1 u Å f2 364 7 . Three forces. 16. and demonstrate that the angle between the resultant and each of the other two forces is 60°. f 1 and f 2 . Two forces. A mass of 5 kg is suspended by two strings. ! ! 20. b. The forces f 1 and f 2 act in an opposite direction to each ! ! ! ! other. f4 17. Four forces. Calculate the magnitude of the equilibrant of these three forces. C a. If the angle between 3 4 1 ! and f 3 is 45°. determine the resultant of these four forces. 14. Determine the tension in each of the strings. !are acting ! on an object and lie in the same plane. The larger tug pulls twice as hard as the smaller tug. f 1 . f 1 and f 2 . Draw a sketch showing an arrangement of these forces. a. f 3 . with and . each having a magnitude of 1 N. ! ! ! ! 15. ! a. make an angle u with each other when they are placed ! ! ! ! ! ! tail to tail. The! forces f 3 and f 4 also act in ! @ @ @ @ f 35 N f 25 N f opposite directions. PART C 18. Determine the angle between and the resultant. and 10 N act from the corner of a rectangular solid along its three edges. In what direction will the ship move? 19. are arranged to produce equilibrium. and the larger tug is 20° off the starboard bow. from two points that are 40 cm apart and at the same level. ! f b. The magnitude of the ! resultant of these two forces is 25 N. Two tugs are towing a ship. 24 cm and 32 cm long. Explain how to determine the angle between the equilibrant and the other two vectors. .