Lab #:! 1!
Name: Calderón, Pedro I.!
Station: 3 ! Date: !
II.
05 Sep 2012!
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Title: Vector Resolution with Force Tables
Partners: Caylor, W. - Dobbin, J.M. - Ganser, K.
BACKGROUND:
We will be working with vectors.
Vectors are physical representations of phenomena that have both magnitude and
direction, such as velocity, acceleration, or force. In order to properly understand
vectors, we will use a given vector set to first find then add their components (by
using SIN, COS, and TAN first to break down vectors into x- and y-components,
then adding them), determining the resultant theoretically by use of trigonometry
and then proving that physically with force tables by using the restoring vector to
balance the original vectors in the set.
III. CAUTIONS:
General lab safety, and since we are using slotted masses, we will make sure to use
closed-toed shoes.
IV.
PURPOSE:
Given a set of vectors, determine the resultant of the set. In order to test our
resultant, we will use the vector set and the exact opposite of our resultant - the
restoring vector. If our calculations are correct, the restoring vector will cancel out
the effects of the original vectors in the set.
V.
HYPOTHESIS:
Please refer to table in ANALYSIS section.
VI.
EXPERIMENT:
A) VARIABLES:
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Controlled: Force Table with pulley clamps, hooks for slotted masses and
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strings on ring(Factors that could change but remain the same during experiment)
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Independent: The vectors in each vector set -placement and mass, thus
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direction and magnitude (Factors that the experimenter varies intentionally)
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Dependent: The resultant vector and thus the restoring vector. (Factors that
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experimenter expects will change as a result of changing the independent variable)
B) MATERIALS & EQUIPMENT:
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force table, pulley clamps, hooks for slotted masses, slotted masses, ring with
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strings, pin for holding ring before proving, calculator
C) PROCEDURE:
1)Draw out each vector set to get an idea where it is (relative to the origin at 0 °
and in which quadrant) so as to determine (VERY IMPORTANT) if the xcomponent and y-component for each vector in the
set are either positive or negative.
2)Once you have drawn out each vector, select the
smallest possible angle with reference to one of the
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α
four cardinal points, viz. NSEW. For example, if
β
one vector is at 62 ° (“ α” as measured from North),
you could use the reference angle from East as 90 ° to set your angle instead
to 28 ° (“β” since it is 28 ° from positive x-axis, or East, which is 90 °).
3)Remembering that Soh-Cah-Toa tells us that sine relates the side opposite an
angle divided by the hypotenuse and that cosine relates the side adjacent to
an angle divided by the hypotenuse, determine the x-component and ycomponent of each vector in a set.
4)Once done finding components of all the vectors in a set, determine the sum
of the x-components as ∑x and the sum of the y-components as∑y.
5)Find the resultant magnitude as the √ ∑x2 + ∑y2
6)Now draw the larger of the two vectors -paying
special attention to the sum: if negative as down or
left, and if positive as right or up - then draw the
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θ
smaller vector. For example, if I had ∑x with a value !
of –30.0 and ∑y with a value of 16.0, then I would
first draw a vector going left 30.0 units and then going up 16.0 units.
Using math from Step 5, my resultant would thus be 34.0 units.
7)Since we draw the larger vector first, it is the most important one. Therefore,
we use the smaller vector to describe the larger. So using the example from
Step 6, I would say the resultant vector is 34.0 units North (since 16.0
units up) of West (since 30.0 units left).
8)Having found the general direction, we can now find the angle; merely use
the identity that TAN θ is opposite/adjacent. In the example from Step 6
then, our angle θ for out Resultant vector would be 28 ° North of West, or
298 ° as its absolute angle referenced from North (that would be 270 ° for
West plus an additional 28 °) HOWEVER, since I want the RESTORING
vector (which will balance out the original vectors in the set), then I would
have to either add or subtract 180 ° from my resultant vector. Hence, my
RESTORING vector here would be 118 ° .
9)Having thus determined my RESTORING vector, we will now test our
hypothesis. To do so, place each of the vectors in the set at the proper angle,
with the magnitude being proportioned as the equivalent mass in grams.
10)We then do the same by placing our restoring vector. If our calculations are
correct, then the ring will float in the middle of the force table since all
vectors will be in equilibrium.
11)Be sure to record your results in the appropriate data table.
12)If the ring does NOT display equilibrium, then note the failure and go
back to recalculate the set if you have sufficient time.
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VII. ANALYSIS
As indicated in the procedure section, break apart each vector into x- and ycomponents, then add them to find Resultant and determine RESTORING vector.
v1= 150 units, at 0°
Set A
v1x = 0!!
1
4
40°
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v2x = 150 * cos (10°)!!
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= 148! !
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∑x = 148!
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v1y = 150 u
v2y = –150 * sin (10°)
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= –26
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∑y = 124
R = √ 1482 + 1242! = 193 units at 40° S of W or 220°
v2= 150 units, at 100°
I use East and reference angle of 10°
South of East to break apart v2 into
its components
Resultant:
2
RESTORING
The RESTORING vector is 180° from
the Resultant vector (you will either add
or subtract 180° to find the new angle),
you can also reverse the directions
3
vy =124
θ=40°
N of E
vx =148
always draw the larger vector first,
then draw the smaller second
but describe as “smaller of larger”
In this case, we drew vx first, then
the vy, but describe it as “Y of X”
or “North of East”
Set B
v2= 150 units,
at 30°
v1= 150 units,
at 0°
45°
v3= 100 units,
at 225°
v1x = 0!!
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v1y = 150 u
v2x = 150 * sin (30°)! !
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= 125! !
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v2y = 150 * cos (30°)
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= 217
v3x = –100 * cos (45°)!
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= –71! !
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∑x = 54!
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v3y = –100 * sin (45°)
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= –71
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∑y = 296
R = √ 542 + 2962!
Use West and
reference angle
of 45° West of South.
= 301 units at 10° W of S or 190°
vx =54
Resultant:
!Remember to make sure
to keep components
negative or positive
depending on location!
vy =296
RESTORING
θ=10°
E of N
always draw the larger vector first,
then draw the smaller second
but describe as “smaller of larger”
In this case, we drew vy first, then
the vx, but describe it as “X of Y”
or “East of North”
Table 1: Hypothetical vs. Experimental Results
Vector
Set
Hypothetical
Restoring Vector
Results of
Proving
A
193 units at 40° S of W (230°)
Incorrect
B
301 units at 10° W of S (190°)
Correct
C
xxx units at yy° A of B (CCC°)
Correct
Observations or Corrections
Failed to use correct force, fixed & worked
D
*Make your data tables look nice and neat. Always include units, and where necessary, uncertainties.
*If you have a graph, make sure the axes are clearly labeled and that you are correctly scaling your graph...taking a
half-page or whole page for a graph would be fine.
*When reporting the BFL for a graph from a linear regression, ALWAYS include the actual equation as well as R
and R2 values.
*If you are merely repeating measurements for sake of precision and reliability, then always include your mean,
%SD, and where appropriate, % error from either expected results or comparison to literature value.
*As always, be clear on your math and pay attention to SF and DP in calculations.
VIII. CONCLUSION
Evaluation:
(My lab was good, good, oh it was good, oh.. oh..oh-oh-OH!)
As evidenced by the fact that three out of four of our hypothetical restoring vectors
were correct on the first attempt, we can be reasonably sure that our calculations
and procedure were correct. (Be genuine in evaluating your hypotheses. Don’t just let me know if you
were correct but let me know the outcomes of the experiment, especially when you reject your hypotheses.)
Errata:
Although we did not commit any mistakes in the execution of this lab, it is worth
mentioning that when we were about to make our first attempt, we almost failed to
take into consideration the mass of the slotted mass hooks as were were setting up.
Although this was only 5.0 grams, the possibility of error was nonetheless present.
(Errata doesn’t mean errors. Errata means errors AND weaknesses. Didn’t find any mistakes in your execution or
calculations? I can believe that. Didn’t find any weaknesses? No Sir. No Ma’am. There must have been
something, either in my set-up or yours that could have borne improvement. And “human error”? NO! It’s obvious
we don’t have “squid error”, so be specific as to the error you committed. Was it parallax, starting ruler at wrong
point, mixing units, reaction time, measuring incorrect variable, switching measurements, what?)
Emendations:
As detailed above, it is necessary to take into account the mass of the slotted mass
hooks when proving your hypothesis. Other than that and making sure your
calculator is set to degrees rather than radians when working on component vectors,
there were no other emendations. Though the force table may have been
rudimentary, it was nonetheless appropriate and accurate for the purposes of our
experiment. (Tell me what you had to do in order to correct your mistakes. Tell me how you would improve
the data collection, verification of data, or design. The generic “use a machine to eliminate human error” is not
only flawed, but lazy. Be thoughtful.)
Extension:
A possible extension of this lab might be to conduct the trials in three dimensions.
Although we could not think of any practical way to execute such a modification, it
would nonetheless prove a challenge, even if done only via computer simulation.
A possible extension would also be to allow pilots traveling with winds that are not
directly head on or from the tail to have a table of values or a program that might
help them to course correct while in flight. (Again, put some thought into it. “Um...instead of using
two vectors, let’s use three!” Seriously? How is that different? Try to add another dimension to the investigation
(no pun intended) so that we can get differing results. How about considering non-negligible air resistance,
verifying with a different method, or using the same technique to explore further? - but specify! Then again, you
may choose to use what you learned in the lab and actually APPLY it, and if so, how?)