Jeff Bivin -- LZHSJeff Bivin -- LZHS

Vector Applications

Created by: Jeffrey BivinEdited by: Aaron Rogers

jeff.bivin@lz95.org

Last Updated: February 14, 2011

Jeff Bivin -- LZHS

Find the component form of the vector that represents the velocity of an airplane descending at a speed of 150 mph at an angle of 30o below the westward horizontal.

150 mph30o

210o0 0150 cos 210 , sin 210v

3 1150 ,2 2

v

75 3, 75v

Jeff Bivin -- LZHS

35o

An airplane is traveling at a speed of 600 mph with a bearing of N35oW at a fixed altitude and no wind. As the plane crosses the Mississippi river, it encounters a wind blowing with a velocity of 60 mph in the direction of N40oE. What is the resultant speed and direction of the plane?

600 mph

125o

0 0600 cos125 , sin125p

40o50o

60 mph

600 mph

0 0600cos125 , 600sin125p

0 060 cos50 , sin 50w 0 060cos50 , 60sin 50w

0 0 0 0600cos125 60cos50 , 600sin125 60sin 50v p w

305.579, 537.454v

plane wind result

result

60 mph

Jeff Bivin -- LZHS

35o600 mph

125o

40o

60 mph 50o

600 mphplane wind result

305.579, 537.454v

2 2305.579 537.454v

618.252v mph

537.454tan305.579

60.379 180o o

119.621o

29.621oBearing N W

result

Jeff Bivin -- LZHS

Joey is playing tug of war with his two best friends. He is pulling with a force of 60N at 210o. Simon exerts a force of 30N at 45o. Austin exerts a force as well. To achieve equilibrium and thus, not lose the game, what is the force Austin must exert and at what angle?

60 N

210o 0 01: 60 cos 210 , sin 210Joey v

45o0 0

2: 30 cos45 , sin 45Simon v 30 N

3??? : cos , sinAustin v f

1 2 3: 0, 0For equilibrium v v v

Jeff Bivin -- LZHS

60 N

210o 0 01: 60 cos 210 , sin 210Joey v

45o 0 02: 30 cos45 , sin 45Simon v 30 N

3??? : cos , sinAustin v f

1 2 3: 0, 0For equilibrium v v v

: 60cos 210 30cos 45 cos 0o ox component f

: 60sin 210 30sin 45 sin 0o oy component f

30.748 cos 0f cos 30.748f

8.787 sin 0f sin 8.787f

Jeff Bivin -- LZHS

60 N

210o

0 01: 60 cos 210 , sin 210Joey v

45o

0 02: 30 cos 45 , sin 45Simon v

30 N3: cos , sinAustin v f

sin 8.787f cos 30.748f

2 23 cos sin v f f

1 8.787: tan30.748

direction

15.95

2 230.748 8.787

31.98 N

Jeff Bivin -- LZHS

A 10,000 pound object is suspended on a wire tied to two poles. The angle between the horizontal and the wire to the shorter pole is 22o. The angle between the horizontal and the wire to the taller pole is 49o. Find the tension in the cable to each pole.

0 01 1: cos 49 , sin 49tall w f

0 02 2: cos158 , sin158short w f

3: 10000 cos 270 , sin 270o oweight w

1 2 3: 0, 0For equilibrium w w w

10000 lbs

22o 49o

f1f2

158o

Jeff Bivin -- LZHS

0 01 1: cos 49 , sin 49tall w f

0 02 2: cos158 , sin158short w f

3: 10000 cos 270 , sin 270o oweight w

1 2 3: 0, 0For equilibrium w w w

10000 lbs

22o 49o

f1f2

1 2: cos 49 cos158 10000cos 270 0o o ox component f f

1 2: sin 49 sin158 10000sin 270 0o o oy component f f

1 2cos 49 cos158 0 o of f

1 2sin 49 sin158 10000 o of f

0

10000

Jeff Bivin -- LZHS

0 01 1: cos 49 , sin 49tall w f

0 02 2: cos158 , sin158short w f

3: 10000 cos 270 , sin 270o oweight w

10000 lbs

22o 49o

f1f2

1 2cos 49 cos158 0 o of f

1 2sin 49 sin158 10000 o of f

Put in a Matrix and use A-1Bor can use RREF

1 9806.09 .f lbs

2 6938.62 .f lbs

Jeff Bivin -- LZHS

Jeff Bivin -- LZHS

A 500 lb cart sits on a ramp inclined at 23o. What is the force is required to keep the cart from rolling down the ramp?

0cos67500

f

0500 cos67 f

500 lbs

23o

500 lbs

67o

195.366 .lbs fGravityNormal

Vector

f

Jeff Bivin -- LZHS

A 1500 lb cart sits on a ramp inclined at 18o. What is the force is required to keep the cart from rolling down the ramp?

0cos721500

f

01500 cos72 f

1500 lbs

18o

1500 lbs

72o

463.525 .lbs f

f

Jeff Bivin -- LZHS

A mover exerts a force of 55 pounds in a horizontal direction on a crate to move it across the floor for a distance of 20 feet. Find the work done on the crate.

55 lbs

55 20work lbs ft

1100work ft lbs

20 ft.𝑤𝑜𝑟𝑘= 𝒇𝒐𝒓𝒄𝒆 · 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆

Jeff Bivin -- LZHS

A mover exerts a force of 55 pounds at an angle of 45o below the horizontal on a crate to move it across the floor for a distance of 20 feet. Find the work done on the crate.

45o

55 lbs

55 cos 45 , sin 45 20 cos0 , sin 0o o o owork

777.817work ft lbs

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖 𝑠𝑡𝑎𝑛𝑐𝑒

55 cos 45 , sin 45 20 1, 0o owork

55cos 45 20 55sin 45 0o owork

55cos 45 , 55sin 45 20, 0o owork

20 ft.

Jeff Bivin -- LZHS

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

A sled is to be pulled by a rope at a 37o angle above the horizontal with a force of 63 lbs.. Find the amount of work done to move the sled 15 feet.

37o63 lbs

0 0 0 063 cos37 , sin37 15 cos0 , sin 0work

754.711work ft lbs

15 ft.

0 063 cos37 , sin 37 15 1, 0work

0 063cos37 15 63sin 37 0work

0 063cos37 , 63sin 37 15, 0work

Jeff Bivin -- LZHS

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

A mover exerts a horizontal force of 35 pounds on a crate as it is pushed up the ramp that is 40 feet long and inclined at an angle of 13o above the horizontal. Find the work done on the crate.

13o

35 lbs

35 cos0 , sin 0 40 cos13 , sin13o o o owork

1364.118work ft lbs

40 ft.

35,0 40cos13 , 40sin13o owork

35 40cos13 0 40sin13o owork

35 1, 0 40 cos13 , sin13o owork

Jeff Bivin -- LZHS

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

A mover exerts a force of 35 pounds parallel to the ramp on a crate as it is pushed up the ramp that is 40 feet long and inclined at an angle of 13o above the horizontal. Find the work done on the crate.

13o35 lbs

35 40work lbs ft

1400work ft lbs Note: The force is in the same direction of

the movement. Therefore, all the

force is moving the crate.

Jeff Bivin -- LZHS

A mover exerts a force of 35 pounds on a crate at an angle of 20o below the horizontal as it is pushed up a ramp that is 50 feet long and inclined at an angle of 23o above the horizontal. Find the work done on the crate.

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒23o

35 lbs

35 cos 20 , sin 20 50 cos 23 , sin 23o o o owork

1279.869work ft lbs

50 ft.

35cos 20 , 35sin 20 50cos 23 , 50sin 23o o o owork

35cos 20 50cos23 35sin 20 50sin 23o o o owork

20o

Jeff Bivin -- LZHS

Jeff Bivin -- LZHS

The following slides are the same examples previously examined that calculate the work needed to move an object. However, an alternative method is demonstrated.

Jeff Bivin -- LZHS

A mover exerts a force of 55 pounds at an angle of 45o below the horizontal on a crate to move it across the floor for a distance of 20 feet. Find the work done on the crate.

45o

f

55 lbs45o

055 cos 45 20work lbs ft

777.817work ft lbs

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

55cos 45 fo

55 cos 45o f

Jeff Bivin -- LZHS

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

A sled is to be pulled by a rope at a 37o angle above the horizontal with a force of 63 lbs.. Find the amount of work done to move the sled 15 feet.

37o63 lbs

063 cos37 15work lbs ft

754.711work ft lbs

f

63cos37 fo

63 cos37o f

Jeff Bivin -- LZHS

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

A mover exerts a horizontal force of 35 pounds on a crate as it is pushed up the ramp that is 40 feet long and inclined at an angle of 13o above the horizontal. Find the work done on the crate.

13of

35 lbs13o

035 cos13 40work lbs ft

1364.118work ft lbs 35cos13 fo

35 cos13o f

Jeff Bivin -- LZHS

𝑤𝑜𝑟𝑘= 𝑓𝑜𝑟𝑐𝑒 · 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

A mover exerts a force of 35 pounds parallel to the ramp on a crate as it is pushed up the ramp that is 40 feet long and inclined at an angle of 13o above the horizontal. Find the work done on the crate.

13o35 lbs

35 40work lbs ft

1400work ft lbs Note: The force is in the same direction of

the movement. Therefore, all the

force is moving the crate.