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# University physics. Forces review of basic concepts

## 1. University Physics I

ForcesReview of Basic Concepts

## 2. Vectors and Scalars

All physical quantities (e.g. speed and force) aredescribed by a magnitude and a unit.

VECTORS – also need to have their direction specified

examples: displacement, velocity, acceleration, force.

SCALARS – do not have a direction

examples: distance, speed, mass, work, energy.

## 3. Representing Vectors

An arrowed straightline is used.

The arrow indicates

the direction and the

length of the line is

proportional to the

magnitude.

Displacement 50m EAST

Displacement 25m at

45o North of East

## 4. Addition of vectors 1

4Nobject

4N

6N

6N

object

resultant = 10N

object

The original vectors are called COMPONENT vectors.

The final overall vector is called the RESULTANT vector.

4N

6N

6N

object

4N

object

resultant = 2N

object

## 5. Addition of vectors 2

With two vectors acting at anangle to each other:

Draw the first vector.

Draw the second vector with its

tail end on the arrow of the first

vector.

The resultant vector is the line

drawn from the tail of the first

vector to the arrow end of the

second vector.

This method also works with

three or more vectors.

4N

3N

4N

3N

Resultant vector

= 5N

## 6. Resultant of Two Forces

• force: action of one body on another;characterized by its point of application,

magnitude, line of action, and sense.

• Experimental evidence shows that the

combined effect of two forces may be

represented by a single resultant force.

• The resultant is equivalent to the diagonal of

a parallelogram which contains the two

forces in adjacent legs.

• Force is a vector quantity.

2-6

## 7. Addition of Vectors

• Trapezoid rule for vector addition• Triangle rule for vector addition

• Law of cosines,

C

B

C

B

R 2 P 2 Q 2 2 PQ cos B

R P Q

• Law of sines,

sin A sin B sin C

Q

R

A

• Vector addition is commutative,

P Q Q P

• Vector subtraction

2-7

## 8. Addition of Vectors

• Addition of three or more vectors throughrepeated application of the triangle rule

• The polygon rule for the addition of three or

more vectors.

• Vector addition is associative,

P Q S P Q S P Q S

• Multiplication of a vector by a scalar

2-8

## 9. Resultant of Several Concurrent Forces

• Concurrent forces: set of forces which allpass through the same point.

A set of concurrent forces applied to a

particle may be replaced by a single

resultant force which is the vector sum of the

applied forces.

• Vector force components: two or more force

vectors which, together, have the same effect

as a single force vector.

2-9

## 10. Rectangular Coordinate System

yx

z

I , j , k : Unit Vectors

## 11.

Vector Representation:A Ax i Ay j Az k

Magnitude or Absolute Value:

A A Ax2 Ay2 Az2

## 12. Direction Angles

yA

Ay

y

Az

z

z

x

Ax

x

## 13. Relationships for Direction Angles

Axcos x

A

cos y

Ay

A

Ax

A A A

2

x

2

y

2

z

Ay

A A A

2

x

2

y

Az

cos z

A

2

z

Az

A A A

2

x

2

y

2

z

## 14. Example 1. A force has x, y, and z components of 3, 4, and –12 N, respectively. Express the force as a vector in rectangular coordinates.

F 3i 4 j 12k## 15. Determine the magnitude of the force in previous example:

F 3i 4 j 12kF (3) (4) ( 12)

2

13 N

2

2

## 16. Determine the three direction angles for the force :

Ax 3cos x

0.2308

A 13

x cos 0.2308 76.66 1.338 rad

1

Ay

4

cos y

0.3077

A 13

y cos 0.3077 72.08 1.258 rad

1

## 17.

Az 12cos z

0.9231

A

13

z cos ( 0.9231) 157.4 2.747 rad

1

## 18. Vector Operations to be Considered

• Scalar or Dot Product:• Vector or Cross Product:

• Triple Scalar Product:

A•B

AxB

(AxB)•C

## 19. Consider two vectors A and B oriented in different directions.

BA

## 20. Scalar or Dot Product

Definition:A • B AB cos

Computation:

A • B Ax Bx Ay By Az Bz

Represents the Work done by the Force B during the

displacement A for example.

## 21. First Interpretation of Dot Product: Projection of A on B times the length of B.

(a)A

A

B

A cos

B

## 22. Or alternatively: Projection of B on A times the length of A.

(b)A

A

B cos

B

B

## 23. Some Implications of Dot Product

0The vectors are parallel to each other and

A B AB

90

The vectors are to each other and

A B 0

## 24. Example : Perform several scalar operations on the following vectors:

A 2i 2 j kB 3i 4 j 12k

A A A A

2

x

2

y

2

z

(2) ( 2) (1) 3

2

2

2

B B B B

2

x

2

y

2

z

(3) (4) (12) 13

2

2

2

## 25.

A • B Ax Bx Ay By Az Bz(2)(3) (-2)(4) (1)(12) 10

A • B AB cos

A•B

10

10

cos

0.2564

AB

3 13 39

cos 0.2564 75.14 1.311 rad

1

o

## 26. Vector or Cross Product

The Cross Product of 2 vectors A and B, is a vector Cwhich is perpendicular to both A and B, and whose

Amplitude is (AB sin(θ))

Computation:

Definition:

A × B AB sin un

i

j

k

A × B Ax

Ay

Az

Bx

By

Bz