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This post covers the basic skills you need to know in order to correctly specify the number of significant figures in a calculation. A work sheet is attached for homework. Continue reading Significant Figures: Basic skills→

What are Units of Measure?
What is meant by a System of Units?

Units of measure are names we give to specific items, sizes, lengths, volumes, weights, masses, temperatures, periods of time etc. Most units of measure were developed by scientists and engineers for convenience. The units we use today were agreed on by people across an industry or a science. Some units are related directly to physical phenomenon. These “physical” constants include the speed of light “c”, and the size of an atomic nucleus, the “barn” because scientists were surprised how large the nucleus was when they first measured it and someone jokingly said “It’s as big as a barn!”

Base units are units which cannot be derived from combinations of other units. Derived units are derived from combinations of base units. Base units include meters {m}, kilograms {kg}, seconds {s}, centimeters {cm}, grams {g}, inches {“}, feet {‘}, slugs {} and others which we will discover later in this course. Some derived units are Newtons {N}, Joules {J}, Watts {W}, dynes {D}, and ergs {e}, pounds {#}, and horsepower {hp}. Notice that I frequently but units in {} brackets when defining them.

A system of units is a group of units which are commonly used together. The three most important systems of units are MKS, CGS, and British or Engineering Units.

In this course, we will be using the MKS system {meters, kilograms, seconds} as the MKS system is best suited to describing large objects such as cars, trains, planets, and people. The MKS system was agreed upon and standardized and named “Système international d’unités” or SI for short. The process of standardizing the MKS system of units began in 1791 and was finally agreed to by an international treaty in 1875 by the Treaty of the Metre. The process continued into the 1950’s with the addition of 3 more base units and a large number of derived units.

The MKS or SI System of Units

Here’s a little history of the “meter”, the “kilogram” and the “second”.

The Meter

The idea of the meter as a unit of length started in about 1665 when the Dutch Physicist Christiaan Huygens (arguably one of the greatest minds in the history of science) observed the length of a pendulum which “ticked” with a half period of 1 second (1 second to swing from left to right, another second to swing back), had a length of 39.26 inches. Christian Huygens is also the person who developed the first pendulum clock or “Grandfather Clock” in 1656).

In 1668, Wilkins and Christopher Wren proposed using the length of this pendulum as a standard of length and called the apparatus the “seconds pendulum”. But this definition of the meter did not catch on at the time.

In the 18th century, there were two approaches to the definition of the standard unit of length. One favored Wilkins approach: to define the meter in terms of the length of a pendulum which produced a half-period of one second. The other approach was to define the meter as one ten-millionth (1/10 000 000) of the length of a quadrant along the Earth’s meridian; that is, the distance from the Equator to the North Pole. This means that the quadrant (a section/distance 1⁄4 of the Earth’s circumference) would have been defined as exactly 10 000 000 metres (10 000 km). In 1791, the French Academy of Sciences selected the meridional definition over the pendular definition because the force of gravity varies slightly over the surface of the Earth, which affects the period of a pendulum.

Question: Calculate the radius of the earth based on the meridional definition of the meter.

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The standard meter today is 40.36 inches, or approximately 1 1/3 yards.

Mass

Now that we have the meter as a standard of length, let’s talk about volume. Volume will lead us to a definition of mass.

There is very little information available about how this started, so I’ll tell what I think happened.

In the 1700’s early chemists wanted a standard of volume and mass so they could compare the results of their research. They figured out how to make a square vessel 1/10th of a meter (10 cm, approximately 4 inches) on each side. They probably made the vessel out of slabs of polished stone or glass, and sealed along each edge with wax. Then they agreed that the amount of water contained in this vessel would be a standard of weight or mass. They called the volume of this vessel 1 liter, and they named the weight or mass of the water contained in this vessel 1 kilogram. In 1889, the CGPM (Conférence Générale des Poids et Mesures) created at standard of mass made from a Platinum – Iridium alloy (a very hard allow which could be accurately machined).

Later in 1901 the CGPM clarified that this kilogram was the standard unit mass, resolving the confusion created by using weight and mass interchangeably.

This lesson is the first lesson in a 2 semester high school physics course. In this introduction to Physics, we discuss the main areas of Physics, subtopics and why Physics is important.

Learning Objective: to understand the meaning of significant figures, to be able to calculate numbers to the correct significance, and to understand how significant figures relate to the precision of a measurement.

Precision: I asked my students how old they were and their answers ranged from 15 to 16. One student said 15 and a half, another said “I’m almost 16”. Much to my surprise, no one answered “I’m 15 years, 2 months, 4 days, 17 hours, 18 minutes and 43 – no wait – 44 seconds old”! The question to ask is how large or small a quantity do you need to properly represent something. In most conversations, saying you are 15 or 16 is good enough. We could say that you approximate your age to the last full year. So the degree of precision is 1 year. You’re either 15 or 16 – that’s it. To be more precise, you might say that you are 15 and 1/2. Then the degree of precision would be 1/2 year. You can keep getting more and more precise to months, days, hours and seconds or even to 1/1000ths of a second. This would be a very high degree of precision to be sure.

How does precision affect mathematical calculations?

The least precise measurement or number determines the overall precision of a calculation.

Consider the number 3. Is 3 really the same as 3.0? The answer is that you don’t have enough information to tell for sure. In fact the number 3 actually represents any number from 2.5 to 3.4999… which has been rounded to the nearest integer. 2.5 rounds up to 3 as does 2.6, 2.7, 2.8, and 2.9. 3.1 rounds down to 3 as does 3.2, 3.3 etc. right up through 3.4.

Now for a more difficult problem. Is 3×3 the same as 9?

The smallest possible value for 3 could be 2.5. 2.5×2.5 = 6.25 rounds down to 6. The maximum value of 3 could be 3.4999. 3.4999 x3.4999 = 12. Take the average of these results (6 + 12) / 2 = 18/2 =9.

Learning objective: To understand powers of 10, know how to multiply and divide powers of 10, and know how to convert from decimal to scientific and engineering notation.

Part 1: What are Orders of Magnitude and powers of 10?
So you’re sitting at coffee shop and some Physicists are writing order of magnitude calculations on the back of an envelope. You want to listen in and comment on the discussion but first you need to understand what they’re talking about. This post is a simple study guide to help you understand and use powers of 10 and orders of magnitude.

First some simple definitions;

Decimal notation: 3.14159 ie numbers and a decimal point. More examples: 73,015.238, 1.059, 0.00063 etc.

Scientific notation: a decimal number between 1.0 and 9.999… multiplied by 10 raised to some power (also called an exponent). scientific notation examples: 6.022×10^23 (Avogadro’s number), 1×10^100 a 1 followed by 100 zeros also known as 1 google, 1.602×10^-19 the charge of an electron, 6.67×10-11 the gravitational constant………. and the list goes on. Notice that the carrot “^” in 10^23 means 10 is raised to the 23rd power. This is the same as saying 1 followed by 23 zeros. Formally, the correct way to write a number in scientific notation is 10 with the exponent written as a superscript but this is less common today because it’s easier to type ^ than to make a superscript.

Exponential notation: same as scientific notation

Engineering notation: Engineers started typing equations a long time ago, there was no “^” key on most of the old typewriters so engineers simply used the letter “e” or “E” to mean exponential notation. For example, 6.022×10^23 could be written as 6.022e23 or as 6.022E23. Negative exponents work the same way so 1.062×10^-19 = 1.062e-19 = 1.062E-19.

Order of magnitude: the exponent only, note round the number part down to 1 or up to 10 and adjust the exponent. examples: 1.062×10^-19, 1.062 rounds down to 1×10^-19 so the order of magnitude is -19 while 6.022×10^23 rounds up to 10×10^23 = 1×10^24 has an order of magnitude of 24.

Simple! Right?

But what do these numbers – powers of 10 or orders of magnitude really mean?

Let’s watch the following 1-1/2 minute video to see how these numbers relate to the real world.

Here’s another, more detailed video with explanations. Please watch this video for homework. Answer all the questions following the video.

Questions:1] What is the order of magnitude of the size of;
a] our galaxy
b] the Earth
c] a plant cell
d] the width of a DNA molecule
e] an atom
f] the nucleus of an atom

Part 2: How to calculate using powers of 10

Multiplication is really simple. All you do is add the exponents.
examples;
10^6 x 10^7 = 10^13 (add exponents 6 + 7 = 13)
For negative exponents you still just add but don’t lose the minus signs.
example: 10^4 x 10^-6 = 10^-2 (add exponents 4 + (-6) = 4-6 = -2)

Dividing works the same way except that your bring the power of 10 in the denominator up to the numerator and change its sign. Formally, this is the same as multiplying top and bottom with the power of 10 with its sign changed (reciprocal).

example: (10^4)/(10^6) = (10^4)/(10^6) x (10^-6)/(10^-6)

= (10^4 x 10-6)/(10^6 x 10^-6)

numerator: (10^4 x 10-6) = 10^-2

denominator: (10^6 x 10^-6) = 10^(6-6) = 10^0 = 1

So the answer is 10^-2 – in decimal notation this is 0.01