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Opus in profectus

Vector Resolution & Components

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Problems

practice

  1. A laser beam is aimed 15.95° above the horizontal at a mirror 11,648 m away. It glances off the mirror and continues for an additional 8570. m at 11.44° above the horizon until it hits its target. What is the resultant displacement of the beam to the target?
  2. Three forces act on a point: 3 N at 0°, 4 N at 90°, and 5 N at 217°. What is the net force?
  3. A cyclist heads due west on a straight road. The wind is blowing from 248° at 10 m/s.
    1. Is this wind more like a headwind or a tailwind?
    2. What is the headwind/tailwind speed?
    3. What is the crosswind speed?
  4. One unfortunate winter day I happened to slip on an icy ramp inclined 37° to the horizontal. Find my acceleration down the ramp given that the acceleration due to gravity points straight down and has a value of 9.8 m/s2. (Assume the ice is perfectly frictionless.)

conceptual

  1. Prove that the angle between the surface of an inclined plane and a the horizontal (the angle of inclination) is equal to the angle between the normal of the plane and the vertical.

numerical

  1. A utility pole is held upright with a single guy wire under 3,500 N of tension. The guy wire meets the ground at a 60° angle. What is the magnitude of the component of this force that…
    1. prevents the pole from tipping over?
    2. keeps the pole in contact with the Earth?
  2. A force of 90 N is applied to a push lawn mower along a handle that makes a 30° angle with the vertical. What is the magnitude of the component of this force that…
    1. propels the lawnmower?
    2. keeps the lawnmower in contact with the ground?
  3. A boy pulls a sled along level ground by means of a rope under 78 N of tension. The rope makes an angle of 37° with the horizontal. What is the magnitude of the component of the tension that…
    1. actually moves the sled?
    2. is essentially wasted?
  4. A cyclist heads out on a straight road with a bearing of 60° east of north. A north wind is blowing at 10 m/s.
    1. Is this wind more like a headwind or a tailwind?
    2. What is the effective speed of this headwind/tailwind?
  5. A cyclist heads out on a straight road with a bearing of 60° east of north. A southwest wind is blowing at 10 m/s.
    1. Is this wind more like a headwind or a tailwind?
    2. What is the effective speed of this headwind/tailwind?
  6. Three of the four busiest airports in the US are Hartsfield-Jackson in Atlanta, Georgia (ATL), O'Hare in Chicago, Illinois (ORD), and Dallas/Fort Worth in Texas (DFW). Complete the following table for roundtrip airplane journey from Atlanta to Chicago to Dallas to Atlanta. Note: Headings are measured clockwise from due north.
    A roundtrip airplane journey
    departure arrival distance heading
    ATL ORD 975 km 342.8°
    ORD DFW 1291 km 221.6°
    DFW ATL  
    round trip   n/a

worksheets

  1. worksheet-components.pdf
    Calculate the magnitude (in centimeters) and direction (in standard form) of the resultant vector for each diagram using the method of components. Use the 1 cm grid lines towards this end. A ruler and protractor are not needed for this exercise.