__Experiment 2__

The objective is to (1) practice the polygon
method of vector addition, and (2) compare
the graphical results with calculation (analytical solution) to get an idea of
how accurate the graphical method used is**.**

__Equipment:__

A protractor, a Metric ruler, and a few sheets of graphing paper

__Theory:__

The resultant of
two or more vectors is a vector that is equivalent in its physical effect to the
action of the original vectors**.** For example, if three force vectors were
acting on an object, these three forces could be replaced by a single vector
(the resultant), and the object would experience the same net effect**.**

**Note:** In the following sections, "gf" means gram-force**.** 1gf is
the force of gravity on the mass of one gram**.**

**Finding the resultant of three forces by
the** **polygon
method:**

**Given:** **A** =
20**0**gf at 0**.**0** ^{o}**,

As shown in the example above, draw a polygon with
the three given vectors A,
B, C by
placing the vectors, one after another, on a tail-to-tip basis**.**
First draw A.
Then from the tip of A, draw B.
Next, from the tip of B, drawC.
Finally, connect
the tail of A (
the first one) to the tip of C (the
last one) to obtain the resultant**.**
The resultant is formed by a vector drawn from the tail of the first vector to
the tip of the last vector**.** Find the gram-force equivalent of the length
of the vector**.** That is the magnitude of the resultant**.** The angle
that it makes with the positive x-axis is its direction**.** Measure it with
the protractor**.** Choose a
scale large enough to make the drawing cover almost the whole sheet of the
graphing paper**.** Make sure not to change the original directions of each
vector as you complete the polygon step-by-step**.**

__Finding the resultant of the same three forces by
the analytical
method:__

__Same
Example__:

** **
**Given:** **A** =
20**0**gf at 0**.**0** ^{o}**,

The x- and y-components of the vectors are:

A_{x} =
20**0** cos 0**.**00˚ = 20**0** gf
Ay = 20**0** sin 0**.**00˚ = 0**.**00 gf

B_{x }=
15**0** cos 35**.**0˚ =123 gf
B_{y} = 15**0** sin
35**.**0˚ = 86**.**0 gf

C_{x}= 25**0** cos
13**0**˚ =
-161 gf
C_{y} = 25**0** sin
13**0**˚ =
192 gf

**R _{x} **=
A

**R _{y} **=
A

**R = R _{ABC} =**

**Step 1:** Calculate
the horizontal and vertical components of each force **A**, **B**,
and **C.**

**Step 2:** Sum the components in
the** x-**direction to obtain **R _{x}.**

** **
**Step 3: **Sum the components in the **y-**direction
to obtain **R _{y}.**

** ****Step 4:** Compute the
magnitude and direction of the resultant using

** ****Step
5:** Draw a sketch of **R _{x}** and

__Procedure:__

The data for this experiment are the three vectors
(**A, B, and C**), as "Given" the **Table
2** below**.**

The purpose is to find the following resultants**:** R_{1},
R_{2}, and R_{3} (one
at a time) using the polygon
method as shown in **Table
1**.

Table 1

R_{1} =
A + B |
R_{2} = A
+ B
+ C |
R_{3} =
A + B - C |

To find each of the resultants R_{1},_{ }R_{2},_{ }and_{ }R_{3} take
the following steps_{ }: ** **

1) Choose a
reasonable scale that gives you a drawing that is big enough for a good
precision and at the same time it does not go out of page**.** Indicate the
selected scale somewhere on the drawing**. For these vectors, if your x-axis
is 1 inch above the lower edge of the paper and your y-axis is 1 inch from the
left edge of the paper, none of **R_{1},_{ }R_{2},_{ }and_{ }R_{3 }**will
go out of page provided that you choose your scale as 1cm = 2N.**

** **2) Add
the vectors by the polygon
method to find the resultant**.**
Record the magnitude and direction of the resultant (that you measure by your
ruler-protractor set) in the **Table
3** shown below**.**
These are your measured
values**.**

3) Solve
for the same resultant as you found in **Step
2**, but
this time by using the analytical method (by calculation and use of
trigonometry)**.** Calculate the magnitude and direction of the resultant
and record it under calculated (accepted) values in the **Table**** 3****.**

4) Calculate a
%error on the magnitude and a %error on the direction and record them in the
space provided in the **Table 3.**
The following %error formula is the one we will be using in this lab throughout
the semester**. **Note: In
your report, the percent error formula must be shown under "
The Comparison of the Results" section**.** You
simply show this formula and only the calculated values of %errors**.** It
is not necessary to show the repeated substitution of numbers in the %error
formula.

__Data:__

Table 2

Vector | Magnitude | Direction |

A | 25.0N |
35.0^{o} |

B | 10.0N |
120.0^{o} |

C | 15.0N |
155.0^{o} |

Table 3

Resultant |
Measured |
Calculated (Accepted) |
%error on Magnitude | %error on Direction | ||

Magnitude (N) | Angle (^{o}) |
Magnitude (N) | Angle ( ^{o} ) |
|||

R1=A+ B |
||||||

R2 =A+
B+C |
||||||

R3 =A+B-C |

Show sample calculation, for example, the complete calculation for R_{1}.

__Comparison
of the results:__

Provide the percent error formula used as well as the calculation of the
percent errors**.**

** Conclusion:**

State your conclusions of the experiment**.**

__Discussion:__

Provide a discussion if necessary**.**

__Questions:__

Which method is more precise, graphical or analytical method?

Why is the polygon method generally considered to be the most reasonable graphical technique?