Vector Resolution & Components

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

prev | up | next

---

Discussion

orthonormal coordinates

• An electro-optical computer mouse has two choppers mounted at right angles.
• The vestibular system, located inside the human skull behind each ear, consists of three mutually perpendicular, angular acceleration detectors called semicircular canals. The utricle senses motion in the horizontal plane while the saccule senses acceleration in the vertical plane. What's the third? (Alcohol consumption changes the density of the fluids in these canals which leads to the sensation that the room is spinning.)

Inside an electro-optical mouse.
	x chopper
	y chopper

components in a rotated coordinate system, the inclined plane

conversion between alternate coordinate systems: polar and rectangular.

Summary

• bullet

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. What is the resultant displacement of the beam?

   Table …

   	magnitude	direction	x-component	y-component
   first beam	11,648 m	15.95°	11,200 m	3,200 m
   second beam	8570. m	11.44°	8,400 m	1,700 m
   resultant	20,200 m	14.04°	19,600 m	4,900 m

   Drawing …

   [magnify]

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?

   Solution …

   Resolve the vectors into their components along the x and y axes. (Watch the signs.) Then add the components along each axis to get the components of the resultant. Use these to get the magnitude and direction of the resultant. Problems with a lot of components are easier to work on when the values are written in table form like this …

   	magnitude	direction	x-component	y-component
   first force	3 N	0°	+3 N	0 N
   second force	4 N	90°	0 N	+4 N
   third force	5 N	217°	−4 N	−3 N
   resultant	1.4 N	135°	−1 N	+1 N

   Drawing the situation is also a pretty good idea.

   [animate]

3. A cyclist heads due west on a straight road. A northeast 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?
   Solutions …
   1. Answer it.
   2. Answer it.
4. One unfortunate winter day I happened to slip on an icy ramp inclined 30° 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.)
   • Answer it.

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 lawnmower 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 with which he pulls the sled makes an angle of 37° with the horizontal. What is the magnitude of the component of the tension that …
   1. actiually moves the sled?
   2. is 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?

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 gridlines towards this end. A ruler and protractor are not needed for this exercise.
2. worksheet-words.pdf
   A copy of all the word problems in this section are also available on a pdf worksheet.

Resources

• general
  • The Mechanical Universe and Beyond (video on demand, login required)
    • Vectors, Physics must explain not only why and how much, but also where and which way.

---

prev | up | next

Another quality webpage by Glenn Elert

home | contact bent | chaos | eworld | facts | physics