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EXPERIMENT #4 - FORCE VECTORS THEORY: Graphical Vector Addition Vectors are represented graphically by arrows that originate from a point of...

complete the graphical part show your calculationsEXPERIMENT #

4

â€“

FORCE VECTORS

THEORY:

Graphical Vector Addition

Vectors are represented graphically by arrows that originate from a point of reference in

a mathematical coordinate system. The length of a vector arrow (drawn to scale on graph

paper) is directly proportional to the magnitude of the vector, and the arrow points in the same

direction as the vector itself.

The length scale is arbitrary and usually selected for convenience so that the vector

graph fits nicely on a sheet of graph paper. An example of a length scale for a force vector

would be

10

à µÂ±

=

1

à µ

. That is, each ten centimeters of vector length represents

1

Newton of

force. The scaling factor in this case, in terms of force per unit length, is

.

1

à µ

/

à µÂ±

.

In order to determine the vector summation of

à µ

+

à µ

first graphically create a

parallelogram of which

à µ

and

à µ

are adjacent sides (see

FIGUREâ€™ 1

below). The diagonal of this

parallelogram labeled as the resultant vector

à µ

, or simply the vector sum of

à µ

+

à µ

; or by vector

addition

à µ

=

à µ

+

à µ

. The magnitude of this resultant vector is proportional to the length of the

diagonal arrow itself, and the direction of the resultant vector is that of the diagonal arrow. The

direction of

à µ

is specified as being at an angle

à µ

, relative to the positive horizontal axis.

An equivalent method of finding

à µ

is to place the vectors to be added â€œtip-to-tailâ€ where

the tip of the vector

à µ

is attached directly to the tail of the vector

à µ

as seen below in

FIGUREâ€™ 2

.

Vector arrows may be â€œâ€™movedâ€ so long as they remain pointed in the same direction. The â€œtip-

to-tailâ€ method gives the same result as the parallelogram method outlined above.

Algebraic Vector Addition

The magnitude of the resultant vector

à µ

, given by the symbol

à µ

, can be computed if we

are able to calculate the components of the vectors

à µ

and

à µ

. Each individual vector may be

resolved into

à µ

- and

à µ

- components, as depicted below in

FIGUREâ€™ 3

. That is, the vector

à µ

is the

specified by the components

à µ

!

and

à µ

!

in the form

à µ

=

à µ

!

à µ

+

à µ

!

à µ

where

à µ

!

=

à µ

cos

à µ

and

à µ

!

=

à µ

sin

à µ

. The magnitude of

à µ

is given by the hypotenuse of a right triangle such that

à µ

=

à µ

!

!

+

à µ

!

!

which is the Pythagorean Theorem. Using trigonometry we also have the equation

tan

à µ

=

à µ

!

à µ

!

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