Measuring to the Correct number of Significant Digits

1
Measuring to the Correct number of Significant
Digits

• The digits measured depends upon the calibration
  of the instrument

2

• Rules for Mathematical Operations using
  Significant Digits
• The multiplication and division rule The answer
  may contain only as many significant digits and
  the measurement containing the least number of
  significant digits.
• Examples
• (1.13 m) (5.126122 m)
• Calculator display shows 5.78251786 m2
• Correct answer 5.78 m2
• (0.500 cm) (0.200 cm) / (0.005 s)
• 20 cm2/s (only 1 s.d)
• (156 cm) (202 cm) (0.0050 cm)
• calculator display shows 157.56 cm3
• correct answer is 160 cm3 or 1.6.102 cm3
• 0.500 cm m km mi
  
• 100 cm 1000 m
  1.61 km
• Calculator shows 3.1056.10-6 mi

3

• Rules for Mathematical Operations using
  Significant Digits
• Addition and subtraction rule the answer may
  contain only as many decimal places as the
  measurement containing the least number of
  decimal places
• Examples
• 677.6 cm
• 39 cm
• 6.232 cm
• calculator shows 722.832 cm
• correct answer 723 cm
• b. 124.5 g
• 121.5 g
• calculator shows 3 g
• Correct answer 3.0 g

4
Describe the value in scientific notation

• 365,611
• 3.66.105
• 0.0000463
• 4.63.10-5
• Express the value in normal notation to 3
  significant figures
• 1.055.106
• 1055000 1060000
• 4.553.10-3
• 0.00455
• Express the value in normal scientific notation
  to 3 significant figures
• 450.04.10-4
• 4.50.10-2
• 0.000456.10-7
• 4.56.10-11

5
Physics I Basic Trigonometry Functions
sin ? o/h cos ? a/h tan ? o/a ? sin-1
(o/h) ? cos -1 (a/h) ? tan-1 (o/a)
h
o
h2 a2 o2
?
a
See Example 4 pg 7 Unlike the examples,
be sure to solve algebra problems down, not
across as shown always carry at least 2 extra
digits from a calculation and underline them if
applicable
See Example 5 pg 7
tan ? ho / ha
tan ? ho / ha
tan ? ho / ha
ho ha (tan ?)
ho 22.0 m (tan 9.1302o )
ho 67.2 m (tan 50.0o)
? tan-1 (ho / ha)
? tan-1 (2.25 m / 14.0 m)
ho 3.54 m
ho 80.1 m
? 9.1302o
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Factor Label Conversion
Factor-Label Conversions Conversion Factors to
Memorize! English to Metric Factors with 3
significant digits 1.61 km 1.00 mile 454 g
1.00 lb 1.06 qt 1.00 L 1.00 dm3
Factors with infinite significant digits
100 cm 1 m 1000 mm 1 m 1000 m
1 km

• Convert 150 cm to m
• 1. Start with a question mark and the unit you
  are seeking.
• 2. Write the measurement you are given.
• 3. Place the unit on what you are given on the
  bottom of the next conversion factor and continue
  until you reach desired unit

? m
150 cm
__m___ 100 cm
1.5 m
7

• Convert 4.00 gallons to mL
• ? mL

4.00 gal
4 qt
L
1000 mL
1.51.104 ml

gal
qt
L
1.06
Convert 8.00 in/s to m/hr
mi

732 m / hr

in
5280 ft
100. ft2 to cm2
9.29.104 cm2
(2.54 cm)2 in2
100. ft2 .
(12 in)2 . ft2
? cm2
8
Scalar Quantities vs. Vector Quantities

• Scalar Magnitude only
• Magnitude is the size or amount of the given
  quantity
• Example The speed of the car is 45 m/s
• The car has traveled a distance of 5 miles
• Vector Magnitude and Direction
• Example The velocity of the car is 45 m/s at 45o
  N of E
• The cars displacement is 5 miles
  due East

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Graphing Vectors
45 m/s

• A Vector quantity can be represented
• on a graph.
• Arrows are used to represent vectors
• (the length of the arrow indicates the
  magnitude of the vector quantity).
• The vector sum of 2 or more vectors
• yields the resultant vector.

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Vector Addition Vectors pointing in the same
direction

• The resultant vector is the sum of the individual
  vectors.
• Example
• On a hiking trip you travel 10. miles N on the
  first day,
• 20. miles N the second day and 35 miles N the
  third day.
• What is your vector displacement for the 3 days?

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Vectors are represented by arrows on a
coordinate system
The length of the arrow represents the magnitude
of the vector
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Vector Addition Vectors pointing in opposite
directions

• The resultant vector can be determined from
  difference between the sum of the vectors in one
  direction and the sum of the vectors in the other
  direction.
• This is the same as determining the sum of the
  vectors where vectors pointing in opposing
  directions have opposite signs.
• Example The tires push off of the road with
  27000N of force in one direction while the car
  experiences frictional forces of 11000 N from air
  resistance and 9000 N from the road.
• What is the resultant force acting upon the car?

13
11000 N
27000 N
x

9000 N
14
Vector Addition Vectors at 90o angles

• 1. Align the vectors tail to tail
• 2. Form a rectangle from the 2 vectors
• 3. Draw a line from where the tails connect and
  determine the hypotenuse.

15
ExampleCarl Yazstremski hits a baseball 50.0
m/s due east where it experiences a wind of 5.00
m/s due north. Determine the
resultant velocity of the baseball.
Directional angle tan ? (5.00 m/s) / (50.0
m/s) ? tan-1((5.00 m/s) / (50.0 m/s)) ?
5.7106o Magnitude cos 5.7106 o 50.0 m/s / v v
(50.0 m/s) / cos 5.7106o v 50.2
m/s Resultant Vector 50.2 m/s _at_ 5.71o N of E
Or 50.2 m/s _at_ 84.3o E of N

v
?
E
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Vector Components

• A vector is composed of horizontal (x) and
  vertical (y) components.
• Examples
• A vector of 90. m/s _at_ 30.o N of W has a velocity
  component vector pointing North and a velocity
  component vector pointing West.

cos 30.o Vx / 90.m/s Vx (cos 30.o)(90. m/s)
Vx 78 m/s, W sin 30.o Vy / 90.m/s Vy
(sin 30.o) (90. m/s) Vy 45 m/s, N
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Utilizing Component Vectors for Determination of
Resultant Vectors

• Draw all vectors on a graph tail-to-tail.
• For a multiple Vector System break all vectors
  into horizontal (x) and vertical (y) components.
• Place components into a table of horizontal and
  vertical components.
• From the horizontal components, determine 1
  horizontal vector and from the vertical
  components determine 1 vertical vector.
• Determine the Resultant Vector from the 1
  horizontal and 1 vertical component vectors.

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• Determine the Resultant Vector

N
100. m


E
W
15o
70. m
3
85 m
30.o
S
d
?
tan ? 17.38 m / 47.10 m ? tan-1 (17.38 m
/ 47.10 m) ? 20.25o , N of W
cos 20.25o 47.10 m/ d d 47.10 m / cos
20.25o d 50. m
50. m _at_ 20.o N of W
The resultant vector of the 3 original vectors is