Application of Vector

8/12/2019 Application of Vector

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APPLICATION OF VECTOR

ADDITION

There are four

concurrent cable forces

acting on the bracket.How do you determine

the resultant force acting

on the bracket ?



Addition of Vectors

• Trapezoid rule for vector addition

• Triangle rule for vector addition

B

B

C

C

Q P R

B PQQ P R

cos2222

• Law of cosines,

• Law of sines,

A

C

R

B

Q

A sinsinsin

• Vector addition is commutative,

P QQ P

• Vector subtraction



Sample Problem

The two forces act on a bolt at

A. Determine their resultant.

SOLUTION:

• Trigonometric solution - use the trianglerule for vector addition in conjunction

with the law of cosines and law of sines

to find the resultant.



Sample Problem (cont’)• Trigonometric solution - Apply the triangle rule.

From the Law of Cosines,

155cos N60 N402 N60 N40

cos2

22

222 B PQQ P R

A

A

RQ B A

R

B

Q

A

20

04.15

N73.97

N60155sin

sinsin

sinsin

N73.97 R

From the Law of Sines,

04.35



ADDITION OF SEVERAL VECTORS

• Step 3 is to find the magnitudeand angle of the resultant vector.

• Step 1 is to resolve each forceinto its components

• Step 2 is to add all the x

components together and add all

the y components together. These

two totals become the resultant

vector.



Example of this

process,



You can also represent a 2-D vector with a

magnitude and angle.



EXAMPLE

Given: Three concurrent forces

acting on a bracket.Find: The magnitude and

angle of the resultant

force.

Plan:

a) Resolve the forces in their x-y components.

b) Add the respective components to get the resultant vector.

c) Find magnitude and angle from the resultant components.



EXAMPLE (continued)

F 1 = { 15 sin 40° i + 15 cos 40° j } kN

= { 9.642 i + 11.49 j } kN

F 2 = { -(12/13)26 i + (5/13)26 j } kN

= { -24 i + 10 j } kN

F 3 = { 36 cos 30° i – 36 sin 30° j } kN

= { 31.18 i – 18 j } kN



EXAMPLE (continued)

Summing up all thei

and j

components respectively, we get,F R = { (9.642 – 24 + 31.18) i + (11.49 + 10 – 18) j } kN

= { 16.82 i + 3.49 j } kN

x

y

FR FR = ((16.82)2 + (3.49)2)1/2 = 17.2 kN

= tan-1(3.49/16.82) = 11.7°



Sample Problem

Four forces act on bolt A as shown.Determine the resultant of the force

on the bolt.

SOLUTION:

• Resolve each force into rectangular

components.

• Calculate the magnitude and direction

of the resultant.

• Determine the components of the

resultant by adding the corresponding

force components.





READING QUIZ

1. The subject of mechanics deals with what happens to a body

when ______ is / are applied to it.

A) magnetic field B) heat C) forces

D) neutrons E) lasers

2. ________________ still remains the basis of most of today’s

engineering sciences.

A) Newtonian Mechanics B) Relativistic Mechanics

C) Euclidean Mechanics C) Greek Mechanics



READING QUIZ

3. Which one of the following is a scalar quantity?

A) Force B) Position C) Mass D) Velocity

4. For vector addition you have to use ______ law.

A) Newton’s Second

B) the arithmetic

C) Pascal’s

D) the parallelogram



CONCEPT QUIZ

5. Can you resolve a 2-D vector along two directions, whichare not at 90° to each other?

A) Yes, but not uniquely.

B) No.

C) Yes, uniquely.

6. Can you resolve a 2-D vector along three directions (sayat 0, 60, and 120°)?

A) Yes, but not uniquely.B) No.

C) Yes, uniquely.



ATTENTION QUIZ

7. Resolve F along x and y axes and write it in

vector form. F = { ___________ } N

A) 80 cos (30°) i - 80 sin (30°) j

B) 80 sin (30°) i + 80 cos (30°) j

C) 80 sin (30°) i - 80 cos (30°) j D) 80 cos (30°) i + 80 sin (30°) j

8. Determine the magnitude of the resultant (F 1 + F 2)

force in N when F 1 = { 10 i + 20 j } N and F 2 =

{ 20 i + 20 j } N .

A) 30 N B) 40 N C) 50 N

D) 60 N E) 70 N

30°

xy

F = 80 N



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Application of Vector