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Graphical Solution for Vectors

  1. Represent the following resultant graphically: A man walks 30 km east and then 10 km north.
  2. My home is 6 blocks east and 8 blocks south of the building in which I work. Graphically determine the straight-line distance between these two points.
  3. A car travels 10 km east, 16 km south, 14km east, 6 km north, and 4 km west. Graphically determine the car’s displacement.
  4. A treasure map gives the following directions: “Start at the large tree. Go 80 paces east, then 50 paces north, then 60 paces west, and then 18 paces south.” Graphically determine how far from the tree is the treasure.
  5. Graphically determine the displacement between two points if one lies 20 km west and 16 km north of the other point.
  6. Point B is 300 km from point A and at an angle of 50° north of east. A road goes 40 km east from point A and then ends. Starting at the end of the road, graphically determine how far and at what angle to the east must one travel in order to reach point B?

Mathematical Solution for Resultant Vectors

  1. A 110 N force, north, and a 55 N force, east, act concurrently on point P. What is the magnitude and direction of the resultant force? Of the equilibrant force? Ans: 122.98 N, 63.4° (or 63.4°, north of east, or calculator answer of 63.4°); 122.98 N, 243.4° (or 63.4° south of west, or calculator answer of -116.6°)
  2. A 15 N force and a 20 N force act concurrently on point P. Find the resultant force when a) both forces act east (the angle between them is zero), b) the larger force acts west and the smaller force acts east (the angle between them is 180°), and c) the larger force acts south and the smaller force acts west (the angle between them is 90°). Ans: 35 N, east; 5 N, west; 25 N, 233.1° (or 53.1° south of west or calculator answer of -126.9°)
  3. A boat can move at a speed of 10 mph in still water. The helmsman steers the boat east across a river 20 miles wide in which the current flows south at 4 mph. What is the boat’s velocity? How long does it take to cross the river? How far downstream of its starting point does it land? (Hint: What component gets the boat across the river? What component gets the boat downstream?) Ans: 10.77 mph, 21.8° south of east or 338.2°; 2 h; 8 miles
  4. Find the resultant force of the four vectors acting concurrently on point P: 8 N north, 3 N east, 10 N south, and 8 N west. Ans: 5.39 N, 158.2° (or 21.8° north of west or calculator answer of 158.2°)
  5. A 55 N force acts due west on an object. What single force can be added to produce equilibrium? Ans: 55 N, east

Component Method of Vector Addition

  1. Find the x- and y-components of the following velocity vector: 20 m/s, 30° (or 30° north of east). Ans: (17.3 m/s, 10 m/s) or 17.3 m/s east & 10 m/s north
  2. Find the x- and y-components of the following vector: 6 m/s, 16° north of east or 16°. Ans: (5.77 m/s, 1.65 m/s) or 5.77 m/s, east, & 1.65 m/s, north
  3. Find the x- and y-components of a force of 400 N, 125°. Ans: (-229 N, 328 N) or 229 N, west and 328 N, north
  4. A lawnmower handle makes a 40° angle with the ground. A 120 N force is exerted along the handle. Find the force component that tends to push the mower across the lawn and the force component that tends to push the mower down into the lawn. Ans: 91.93 N; 77.13 N
  5. A student pulls a 5 kg box across a floor by means of a rope making a 60° with the floor. He applies a 45 N force along the rope. What is the magnitude of his force that moves the box across the floor? What is the magnitude of the normal force? (Hint: remember to take into accoutn all the vertical forces) Ans: 22.5 N; 10.03 N
  6. An electric sign weighing 1000 N hangs on the end of a horizontal bar extending horizontally outward from a wall. The end of the bar is connected to the wall by a support wire that makes an angle of 35° with the bar. What is the thrust in the bar and the tension in the wire? Ans: thrust=1428 N; tension=1743.45 N
  7. A window pole is 3m long and is held with its upper end hooked onto a vertical window mounted on a wall. The lower end of the pole is 0.6 m horizontally from the wall. A pull of 180 N is exerted along the pole. Find the horizontal and vertical components of the force exerted on the pole. (Hint: Do you have enough information to determine the angle? Draw a sketch, labeling all forces.) Ans: horizontal=36 N; vertical=176.36 N
  8. Find the resultant vector of two forces acting simultaneously at point P: 100 N, 140° & 60 N, 50°. Ans: 116.62 N, 109.03°
  9. Find the resultant vector: 80 N, 0°, 100 N, 45°, 110 N, 150°, and 160 N, 200°. Ans: 118.51 N, 143.2°

Vectors Notes

Vectors Sample Problems