__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 effects 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
their 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**°**,
**B** = 15**0**gf at 35**.**0**°**,
and **C** = 25**0** gf at 13**0****°**

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, draw
C. 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**.**
Again, 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**°**,
**B** = 15**0**gf at 35**.**0**°**,
and **C** = 25**0** gf at 13**0****°**

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 big enough for precision measurement
and at the same time the drawing does not go out of page**.** Indicate
the selected scale on the drawing**. For these vectors, if your x-axis is
about 1 - 1½ inch above the lower edge of
the paper and your y-axis is about 1 - 1½
inch to the right 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 magnitude and a %error on 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° |

B | 10.0N |
120.0° |

C | 15.0N |
155.0° |

Table 3

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

Magnitude (N) | Angle (°) | Magnitude (N) | Angle ( ° ) | |||

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 the most precise, graphical or analytical method?

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