Grade 12
PHYSICS 1
Module 1: Units Of Measurement
Vectors
1st Semester, S.Y. 2020-2021
Prepared by:
DALE G. DAŇAS
Subject Teacher
________________________________________________________________________________
MDM-Sagay College, Inc.
Office: Feliza Bldg., Marañon St. Pob 2, Sagay City
Campus: National Highway, Poblacion 2, Sagay City, Negros Occidental
Tel.# 488-0531/ email: mdm_sagay2000@gmail.com.
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Module 1: Units of measurement & vectors
Lesson 1. (Conversion of units of measurement in scientific notation )
Learning Outcomes: At the end of the lesson, the student should be able to:
1. Convert units of measurement in scientific notation
What I know
Activity 1:
1.
2.
3.
4.
5.
Answer the following
4 x 1,000 =
45 x 1,000,000 =
3 / 1,000=
55 / 100 =
567 / 10,000 =
_________________
_________________
_________________
_________________
_________________
What is It
Lesson 1 (Conversion of units into scientific notation)
Physical quantity
Imagine you had to make curtains and needed to buy material. The shop assistant would
need to know how much material was required. Telling her you need material 2 wide and 6 long
would be insufficient| you have to specify the unit (i.e. 2 meters wide and 6 meters long). Without
the unit the information is incomplete and the shop assistant would have to guess. If you were
making curtains for a doll's house the dimensions might be 2 centimeters wide and 6 centimeters
long!
Based quantity
Length
mass
Time
Electric current
Temperature
Luminous intensity
Name
Symbol
Meter
m
Gram
g
Second
s
Ampere
A
Celsius
C
Candela
cd
Table 1.1 SI Base Units
All physical quantities have units (e.g. time and temperature)
SI Units (Systeme International d'Unities)
Length is the measurement of distance which could be in meters, centimeters kilometers
etc… every based units has a prefix which will help us understand the true value of the quantity.
These prefixes are use in expressing the values in which help reduce the size of larger or smaller
values of quantity.
Refer to the table for the unit prefixes and its values
Prefix
deca
hecto
kilo
mega
Symbol
da
h
k
M
Value
( in exponent )
101
102
103
106
Prefix
deci
centi
millli
micro
1
Symbol
d
c
m
μ
Value
(in exponent)
10-1
10-2
10-3
10-6
giga
tera
peta
exa
zetta
yotta
G
T
P
E
Z
Y
109
1012
1015
1018
1021
1024
nano
pico
femto
atto
zepto
yocto
Table 1.2: unit Prefixes
n
p
f
a
z
y
10-9
10-12
10-15
10-18
10-21
10-24
Scientific notation is a way of writing very large or very small numbers. A number is written
in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For
example, 650,000,000 can be written in scientific notation as 6.5 ✕ 108
Let’s take for example 15mg
15 mg is the value with prefixes; this is also equal to 15x10-3 grams, in decimal this would be
equal to 0.015 (g) grams and the scientific notation of this value is
1.5 x 101 mg (milligram) ----------- in terms of milligrams
1.5 x 10 -2 g (grams). --------------- in terms of grams both have the same value
But how???
Step1.
First you convert the given units into the desired unit of measurement in this case milligram into
gram, so the 15mg must be divided by 1000 in order to convert it into grams so 15/1000 =
0.015g ….. we know that 1g = 1000mg
this is the process
15mg ( 1g ) =
15gram = 0.015gram
1000mg
1000
Note: notice that mg is at the bottom of the fraction in order to cancel the unit mg and remains the unit
grams. if ever the you convert grams to milligrams the fraction would be reversed e.g. 15g is equal
15,000milligram
15g(1000mg) = 15,000mg
1g
Step 2
You have to follow the format of the scientific notation by moving the decimal point either left of
right in order to produce a number higher than 1 and lower than 10. In this case you have to
move to the right 2 times
0.015g = 1.5 x 10 -2grams
Note: counting decimal point to the right gives you x10 with negative exponent
Counting decimal to the left gives you x10 with positive exponent
What if the value is 15,000,000,000 grams? How would you write it in scientific notation in terms
of kilograms?
Step 1. Convert grams to kilograms …. We know that 1kg = 1000g
15,000,000,000 grams ( 1kg) = 15,000,000,000 kg = 15,000,000kg
2
1000grams
1000
Step 2.
Notice that it’s not a decimal value which means that the point is on the far right after the last 0
which means we have to count going to the left, in this case its 7 places to the left in order to
produce 1.5 x 107 kg
Note : if the given unit is the same as the desired unit no need for conversion , you just have to move the
decimal point either to the right or left e.g. 100,000mm is equal to 1x105 mm
What I have Learned
Activity 2
Write the following values with prefixes into scientific notations
1. 250,000 mg (milligram)
___________________g (grams)
2. 50000 km(kilometer)
___________________m (meter)
3. 0.005 cm (centimeter)
___________________cm (centimeter)
4. 500 kg (kilogram)
___________________g (grams)
5. 400 Mw(megawatts)
___________________w (watts)
2 types of measurement
 SI units ( metric system)
 Imperial units ( English units of measurement)
SI units
Meters (m)
Centimeters(cm)
Millimeter (mm)
Gram (g)
Liter (L)
Imperial units
Yards (yd)
Feet (ft)
Inches (in)
Pound (lb)
Ounces( oz)
Table 1.3 units Counterpart
SI Units
1 meter(m)
2.54centimeter(cm)
1 kilogram(kg)
1.6 kilometer(km)
1 liter(L)
Imperial units
3.28 feet(ft)
1 inch (in)
2.2 pounds(lb)
1 mile(mi)
33.8 ounce (oz
Table1.4 unit conversions
Converting units from English system units to Metric system units
Examples:
10 meters (m) would be equal to 32.8 feet (ft)
45 feet (ft) would be equal to 13. 72 meters (m)
BUT HOW???
How to Change Units| the \Multiply by 1" Technique
Firstly you obviously need some relationship between the two units that you wish to
convert between. Let us demonstrate with a simple example.
We know that 1 m = 3.28 ft
10 meters (3.28 ft) = 32.8 ft
1 meter
(units meters cancels and remains the unit ft)
45 ft (1meter) = 13.72 meter (m) (unit feet cancels and remain the units meter)
3.28 ft.
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What I have Learned
Activity 3
Convert the following examples
1. 30 m =
2. 45 cm =
3. 50 lbs =
4. 60 km =
5. 45 oz =
_________ ft
_________ in
_________ kg
_________ mi
_________ L
Lesson 2: Accuracy and precision of measurement
Learning Outcomes: At the end of the lesson, the student should be able to:
1. Differentiate accuracy from precision
___________________________________________________________________
Accuracy is how close a measured value is to the actual (true) value.
Precision is how close the measured values are to each other
Examples of several values on a number line
Example on a target
How to remember ?
Accurate is correct (a bullseye)
Precise is repeating (hitting the same spot but maybe not the correct spot)
Degree of accuracy
Degree of accuracy depends on the instrument we are measuring with.
As a general rule. The degree of accuracy is half a unit each side of the unit of measure
Examples :
When an instrument measures in "1"s
any value between 6½ and 7½ is measured as "7"
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When an instrument measures in "2"s
any value between 7 and 9 is measured as "8"
(Notice that the arrow points to the same spot,
but the measured values are different!
We can show the error using “ plus – minus sign” ±
When the value could be 6½ and 7½
7 ± 0.5
The error is ± 0.5
When the value could be between 7 and 9
8±1
The error is ± 1
Example: a fence is measured as 12.5 meters long, accurate to 0.1 of a meter
Accurate to 0.1 m means it could be up to 0.05 m either way:
Length = 12.5 ± 0.05
So the height of every single piece of a fence is anywhere between
12.45m and 12.55m
What I have Learned
Activity 4
1. If you’re instrument measures in 2's then any value between which two values would
be measured as "6"?
a. 4 and 8
c. 5 and 7
b. 4.5 and 7.5
d. 5.5 and 6.5
2. If your instrument measure in 1’s then any value between which two values would be
measure as 6?
a. 4 and 8
c. 5 and 7
b. 4.5 and 7.5
d. 5.5 and 6.5
3. Which one shows high precision but low accuracy
4. Which one shows high precision and high accuracy
5. Which one shows low precision but high accuracy
Lesson 3: Errors
Learning Outcomes: At the end of the lesson, the student should be able to:
1. Differentiate random errors and systematic errors
___________________________________________________________________
The Absolute error is the difference between the actual value and measured value. In
the example above the absolute error is 0.05 m because the actual value is 12.5 and the
measured values vary from 12.5±0.05.
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The Relative Error is the absolute error divided by the actual value
The Percentage Error is the relative error multiplied by 100%
Example: fence (continued)
Length = 12.5 ±0.05 m
So:
Absolute Error = 0.05 m
And:
Relative Error = 0.05 m/12.5 m = 0.004
And:
Percentage Error = 0.4%
Another example:
The thermometer measures to the nearest 2 degrees. The temperature was measured as 38° C
Temperature = 38 ±1°
So:
Absolute Error = 1°
And:
Relative Error = 1°/38° = 0.0263...
And:
Percentage Error = 2.63...%
What I have Learned
Activity 5
You measure the plant to be 80 cm high (to the nearest 3cm)
Calculate the following:
Height of the plant(measured value)=
Absolute error =
Relative error =
Percentage Error =
_____________________
_____________________
_____________________
_____________________
Types of Errors
 Systematic errors
 Random errors
Systematic error (also called systematic bias) is consistent, repeatable error associated with
faulty equipment or a flawed experiment design. These errors are usually caused by measuring
instruments that are incorrectly calibrated or are used incorrectly. However, they can creep
into your experiment from many sources, including:
 A worn out instrument For example, a plastic tape measure becomes slightly stretched
over the years, resulting in measurements that are slightly too high
 An incorrectly calibrated instrument like a scale that doesn’t read zero when nothing is
on it,
 A person consistently takes an incorrect measurement For example, they might think the
3/4″ mark on a ruler is the 2/3″ mark.
Random error (also called unsystematic error, system noise or random variation) has no
pattern. One minute your readings might be too small. The next they might be too large. You
can’t predict random error and these errors are usually unavoidable
Random errors do not follow a pattern
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The main differences between these two error types is
Random errors are (like the name suggests) completely random. They
are unpredictable and can’t be replicated by repeating the experiment again
Systematic Errors produce consistent errors, either a fixed amount (like 1 lb) or a
proportion (like 105% of the true value). If you repeat the experiment, you’ll get the same error.
Types of systematic error
1. Offset error is a type of systematic
error where the instrument isn’t set to
zero when you start to weigh items.
For example, a kitchen scale includes
a “tare” button, which sets the scale
and a container to zero before
contents are placed in the container.
This is so the weight of the container
isn’t included in the readings. If the
tare isn’t set properly, all readings will
have offset error
2. Scale Factor Errors these are errors
that are proportional to the true
measurement. For example, a
measuring tape stretched to 101% of
its original size will consistently give
results that are 101% of its true value.
Scale factor errors increase (or decrease) the true
value by a proportion or percentage
Preventing errors
Random errors can be reduced by
 Using an average measurement from a set of measurements or
 Increase the sample size of the set of measurements
Systematic error is difficult to detect and therefore in order to avoid these types of errors you
must
 Know the limitations of your equipment
 Understand how experiment works

What I have Learned
Activity 6:
1. Before the cookies was weighed the reading on the scale was
Then a packet of cookies is weighed and the reading on the scales is became 300 g.
What is the true weight of the packet of cookies?
a. 75g
c. 225g
b. 300g
d. 375g
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2. In question no1 what type of error is it?
a. Offset error
c. scale factor error
b. Random error
d. none from the choices
3. a measuring tape is stretched to 101% of its original size. What type of error it will
produce?
a. Offset error
c. scale factor error
b. Random error
d. none from the choices
4. If the reading on your measuring tool is unpredictable you might have an error that is?
a. Offset error
c. scale factor error
b. Random error
d. none from the choices
5. What is the main difference between random and systematic error?
________________________________________________________
________________________________________________________
Lesson 4: Vector vs. scalar quantity
Learning Outcomes: At the end of the lesson, the student should be able to:
1. Differentiate vector and scalar quantities
___________________________________________________________________
The motion of objects can be described by words. Even a person without a background in
physics has a collection of words that can be used to describe moving objects. Words and
phrases such as going fast, stopped, slowing down, speeding up, and turning provide a
sufficient vocabulary for describing the motion of objects. In physics, we use these words and
many more. We will be expanding upon this vocabulary list with words such
as distance, displacement, speed, velocity, and acceleration. As we will soon see, these words
are associated with mathematical quantities that have strict definitions. The mathematical
quantities that are used to describe the motion of objects can be divided into two categories.
The quantity is either a vector or a scalar. These two categories can be distinguished from one
another by their distinct definitions:


Scalars are quantities that are fully described by a magnitude ( numerical value alone)
Vectors are quantities that are fully described by both a magnitude and a direction.
What I have Learned
Activity 7
Check Your Understanding
1. To test your understanding of this distinction, consider the following quantities listed below.
Categorize each quantity as being either a vector or a scalar.
Quantity
Category
a. 5 m
b. 30 m/sec, East
c. 5 mi., North
d. 20 degrees Celsius
e. 256 bytes
f. 4000 Calories
Lesson 5: Vectors is component form
Learning Outcomes: At the end of the lesson, the student shoul be able to:
1. Rewrite vector in component form
___________________________________________________________________
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Vectors are used to represent a quantity that has both a magnitude and a direction. The vector
is normally visualized in a graph. A vector A could be written as
A
The vectors standard position has its starting point in origin.
The component form of a vector is the ordered pair that describes the changes in the x- and yvalues. In the graph above x1=0, y1=0 and x2=2, y2=5. The ordered pair that describes the
changes is (x2- x1, y2- y1), in our example (2-0, 5-0) or (2,5)..
In our example on the graph the vector ax = 2 and ay = 5 in order to calculate the magnitude of
vector a , we have to use the Pythagorean theorem .
So the magnitude of vector a would be a =√22 + 52
a = =√29 = 5.39
in order to calculate the direction we will use the trigonometric function tangent
tan ϴ = ay / ax = 5/2 = 2.5
ϴ = tan-1 2.5 = 68.20 degrees north of east ..
so our vector a has a magnitude of 5.39 and the direction of 68.2 degrees north of east
note: we can break down the component of a vector either by graphing or by mathematical manipulation
provided the magnitude and angle is given
The most common way is to first break up vectors into x and y parts, like this:
ϴ
The vector a is broken up into
the two components ax and ay
Let’s say vector B is has a magnitude of 36 and the angle of 10 degrees north of east
In order to break down this component of this vector we have to use trigonometric function.
given the hypotenuse and an angle we can use sine function.
ϴ
Remember soh cah toa ???
Sineϴ = opposite / hypotenuse ...now the opposite is the By component of the vector B
cosϴ = adjacent / hypotenuse now the adjacent is equal to the B x of the vector B
solving for By = Sineϴ(hypotenuse)
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By = sin10 ( 36) = 6.25
Solving for Bx = cosϴ(hypotenuse)
Bx= cos10 ( 36) = 35.45
So we can say that the vector B has a component of
By = 6.25 and Bx= 35.45
What I have Learned
Activity 8
Calculate the component of vector x ( note their difference )
1. with a magnitude of 60 with an angle of 35 degrees north of east
2. with a magnitude of 40 with an angle of 90 degrees north
Lesson 6: Adding and Subtracting vectors
Learning Outcomes: At the end of the lesson, the student should be able to:
1. Add and subtract vector quantities
We can combine vectors by adding them; the sum of two vectors is called the resultant vector
Example of adding vector’s component form together
Add the following vectors
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Vector A (7,5) and vector B (3,2)
Note add both x-components and y-component together, in this case there is only two vector if there is
three or more vectors involve the same process applies add all value of x-components and ycomponents
Solution : AB = {7 +(3), 5+2}
AB = (10, 7) ------------this is the resultant vector in component form
Picture below illustrate how true adding vectors together using its x and y components
The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)
Example: add the vectors a = (8, 13) and b = (26, 7)
c=a+b
c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)
Magnitude of a Vector
The magnitude of a vector is shown by two double vertical bars on either side of the vector:
||a||
We use Pythagorean theorem to calculate it:
||a|| = √( x2 + y2 )
Example: what is the magnitude of the vector b = (6, 8) ?
|IbI| = √( 62 + 82) = √( 36+64) = √100 = 10
A vector with magnitude 1 is called a Unit Vector.
What I have Learned
Activity 9
Add the following vectors, calculate the magnitude and determine the direction
1. A(4, 8) and B (3, 10)
2. X( 7,2) , Y(2, 4), Z( -3, 9)
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