Unit 1 Mathematical Toolkit

Physics 12Unit 1 – Mathematical Toolkit

Dr. John LoRoyal Canadian College

2020-2021

1. Measurement and scientific notationsA. Système International d´Unités (SI Units)

• The base units for measurement of fundamental quantities. Other units can be derived by combining these units.

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Quantities Units Symbols

Length Meter m

Mass Kilogram kg

Time Second s

Temperature Kelvin K

Force Newton N

Energy Joule J

Electrical Current Ampere A

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B. Scientific Notation

• Scientific notation is based on exponential notation. In scientific notation, the numerical part of a measurement is expressed as a number between 1 and 10 multiplied by a whole-number power of 10.

• The general form:

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𝑀 × 10𝑛, where 1 ≤ 𝑀 < 10

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C. Metric prefixes

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Prefix Multiple Symbol

tera 1012 T

giga 109 G

mega 106 M

kilo 103 k

hector 102 h

deca (or deka) 101 da

deci 10-1 d

centi 10-2 c

milli 10-3 m

micro 10-6

nano 10-9 n

pico 10-12 p

femto 10-15 f


2. Review of algebra and trigonometry

A. Right-angled triangle:

• An triangle with an angle equal to 90°

• We give names to each side:

➢Adjacent side is next to the angle θ

➢Opposite side is facing the angle θ

➢The longest side is the hypotenuse

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• Trigonometry is good at find a missing side or angle in a triangle.

• For any angle θ:

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sin θ=opposite side

hypotenuse

cos θ=adjacent side

hypotenuse

tan θ=opposite side

adjacent side


B. Pythagorean theorem

• In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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• The formula of the Pythagorean theorem is:

𝑎2+ 𝑏2 = 𝑐2

𝑐 = 𝑎2+ 𝑏2

• If the lengths of any two sides are known the length of the third side can be found.


3. Uncertainty in measurements

A. Precision and accuracy

• The term precision refers to the degree of exactness to which the measurement of a quantity can be reproduced. The precision of an instrument is limited by the smallest division on the measurement scale.

• The term accuracy refers to the extent to which a measured value agrees with the standard value of a quantity. The accuracy of a measurement describes how well the result agrees with an accepted value.


• Example: Describe the following situations.


B. Significant digits

• Significant digits are all the digits that are certain plus a digit that estimates the fraction of the smallest division of the measuring scale.

• Follow these rules to decide if a digit is significant:

a) Nonzero digits are always significant.

b) All final zeros after the decimal point are significant.

c) Zeros between two other significant digits are always significant.

d) Zeros to the right of a whole number are considered to be ambiguous.

e) Zeros to the left of the first non-zero digit in a decimal are not significant.


C. Operations with significant digits

• The sum or difference of two values is as precise as the least precise value.

12.345+ 1.23

13.575 → 13.58

• The number of significant digits in a product or quotient is the number in the factor with the lesser number of significant digits.

12.345x 1.2345

15.2399025 → 15.240


D. Errors

• There are two types of experimental errors that we usually encounter:

a) Random Error results when an estimate is made to obtain the last significant digit for any measurement.


❖ When measuring length, it is necessary to estimate between the marks on the measuring tape. If the marks are 1 cm apart, the random error will be greater and the precision will be less than if the marks are 1 mm apart.

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b) Systematic Error is associated with an inherent problem with the measuring system, such as the presence of an interfering substance, incorrect calibration, or room conditions.


❖ If the balance is not zeroed at the beginning, all measurements will have a systematic error; using a slightly worn meter stick will also introduce error.

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E. Percentage error

• It is defined as

% error =experimental value − accepted value

accepted value× 100%

• When we measured the speed of sound, we found it to be 342 m/s. The accepted values at the same temperature is 352 m/s. What is the percentage error of the measurement?

% error =342 − 352

352× 100% = −2.84%

The negative sign indicates that our measured value was less than the accepted value.


6. Percentage difference

% difference =difference in measurements

average measurement× 100%

• If two measurements of the speed of sound are 342 m/s and 348 m/s. Their % difference is:

% difference =348 − 342

348 + 3422

× 100% = 2.00%


4. Graphing

• A graph is a visual representation of how two quantities are related. Graphing data is more useful than presenting results in a data table.

• A mathematical relationship that shows how these two quantities depend on each other, can be determined from the graph. Many of the laws of science are expressed by such relationships.

• All graphs should include the following information:

➢Title: Description of the relationship illustrated by the graph

➢Axis labels: Given in correct units

➢Scale: number scale used must be consistent; may be different for x and y axes

➢Data: data points and best-fit line


A. Direct proportion (linear relationship)

• A student wishes to investigate the relationship between the volume of a piece of iron and its mass.

• Refer to the data collected in the following table.

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Mass (lb) Volume (cubic ft)

440 1

880 2

1320 3


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0

200

400

600

800

1000

1200

1400

1 2 3

Mas

s o

f ir

on

(lb

)

Volume (cubic ft)

Mass versus volume of Iron

As seen from the data above, doubling the volume doubles the mass; tripling the volume triples the mass. This type of relationship is called a direct relationship.

Unit 1 - Mathematical Toolkits

• Why is it linear?

• It’s because the slope represents density!

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B. Quadratic relationship

• A quadratic graph will have a general shape as follows:



𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 𝑎 𝑥 − 𝑝 2 + 𝑞

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C. Inverse relationship

• An inverse relationship will have a general shape as follows:



𝑦 =𝑘

𝑥𝑛RCC @ 2020/2021

D. Manipulating graphs

• Some relationships can be manipulated mathematically to produce a straight line. This process is called line straightening or linearization.

• In general, a graph that is a quadratic graph, the data can be re-graphed with one of the variables squared.

➢A quadratic graph 𝑦 = 𝑘𝑥2 can be linearized with the x-variable squared; that is, 𝑦 = 𝑘𝑤, where 𝑤 = 𝑥2.

➢A quadratic graph 𝑦2 = 𝑘𝑥 can be linearized with the y-variable squared; that is, 𝑤 = 𝑘𝑥, where 𝑤 = 𝑦2.

• Same strategy can be applied to other types of relationship such as inverse.


• Example: A student performs an experiment to measure how centripetal force changes as the velocity is altered. He collects the data shown on the table below. Using a suitable averaging technique, determine a new set of values that, if graphed, would give a linear graph. Determine the mass of the object if the radius of the circle is 1.15 m.

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Force (N) v (m/s)

20.0 1.0

80.0 2.0

180 3.0

320 4.0

500 5.0 0

100

200

300

400

500

600

1 2 3 4 5

FOR

CE

(N)

VELOCITY (M/S)

F VERSUS V


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Force (N) v (m/s) V2 (m/s)2

20.0 1.0 1.0

80.0 2.0 4.0

180 3.0 9.0

320 4.0 16.0

500 5.0 25.0

0

100

200

300

400

500

600

1 6 11 16 21

Forc

e (

N)

Velocity (m/s)^2

F versus v^2


RationaleCentripetal force on an object undergoing a circular motion is determined by

𝐹𝑐 =𝑚

𝑟𝑣2

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E. Finding slope

• To find the slope of a straight line:

1. Determine the best-fit line (or trendline): This is an educated guess about where a linear equation might fall in a set of data plotted on a scatter plot.

2. Choose two points from the best-fit line; e.g. 𝑥1, 𝑦1 and 𝑥2, 𝑦2 .

3. Put the points into the two-point formula:

𝑚 =𝑦2− 𝑦1𝑥2− 𝑥1


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• Example: Determine the slope of the best-fit line.

• Use the two points: (2, 18) , (4.2, 40)

slope =40 − 18

4.2 − 2= 10 m/s

Note:

1) Don’t use the data points; use the points from the best-fit line.

2) Give correct unit to the final answer.


Unit 1 Mathematical Toolkit

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