The Physics
Hypertextbook
Opus in profectus

# Vector Resolution and Components

## 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 3500 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: Bearing are measured clockwise from due north.
A roundtrip airplane journey
departure arrival distance bearing
ATL ORD 975 km 342.8°
ORD DFW 1291 km 221.6°
DFW ATL
round trip   n/a

### numerical drill

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.
2. Drill
1. Four forces act on an object: 70 N at 0°, 90 N at 90°, 30 N at 180°, and 60 N at 270°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle
2. Four forces act on an object: 90 N at 0°, 70 N at 90°, 60 N at 180°, and 30 N at 270°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle
3. Three forces act on an object: 32 N at 0°, 55 N at 37°, and 24 N at 90°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle
4. Three forces act on an object: 24 N at 0°, 55 N at 53°, and 32 N at 90°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle
5. Three forces act on an object: 55 N at 37°, 32 N at 180°, and 45 N at 270°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle
6. Three forces act on an object: 55 N at 53°, 45 N at 180°, and 32 N at 270°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle
7. Two forces act on an object: 20 N at 37° and 15 N at 127°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle
8. Two forces act on an object: 15 N at −37° and 20 N at +53°. Determine the following quantities (rounded to the nearest whole number)…
1. the net force in the x direction (include the mathematical sign)
2. the net force in the y direction (include the mathematical sign)
3. the magnitude of the net force
4. the direction of the net force as a standard position angle