Moving

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Week 2

The preliminaries out of the way, we began looking at "real" physics in week two — kinematics. Various definitions, as is often the case, were a useful place to start; before we worked through an example problem. After considering the implications of negative values for some of the measurements we're concerned with, we examined the formulæ relating them to each other.

Significant Figures & Rounding

The concept of significant figures, which came up toward the end of the previous post on scientific notation, deserves a bit more of an explanation. when writing a number, particularly a very small or very large number, it is common to write a number of zeros at its start or end, respectively. These leading/following zeros convey no information other than "move along, there's nothing to see here", you might even say that they are rather 'insignificant' when compared to the other digits, which we call significant digits, or significant figures.

One of the attractions of using scientific notation is that it allows us to omit non-significant figures as the only information that they convey (the magnitude of the number — that these places are empty) is given in the exponent.

An important, and related, idea is that of rounding. Much of the mathematics used in physics involves numbers with many of significant digits (the physical constants, for example) or operations which may result in such numbers (like taking roots and the trigonometric functions). When we combine these with other numbers the result will usually have a large number of significant figures, perhaps belying the accuracy of our input numbers. We usually, therefore, round our answers to as many significant figures as the input with the least. Thus our estimate of "3 m or so" does not result in an answer with a dozen digits — near enough it good enough.

For more information on rounding, including the variety of rounding methods, see some of the links below.

More information can be found at the Wikipedia articles on rounding and the Wolfram Mathworld articles on significant digits

Scientific Notation

As the talk of 'prefixes' in the previous post on units suggests, physics involves working with both very large and very small numbers. While we humans are pretty good at working with numbers with rather small numbers of digits (one, two, three, even four or five digit numbers can be handled with a fair degree of precision and reliability), if we increase the number of digits to ten or twelve, then we slow way down and start making more mistakes.

To help reduce the burden of large and small numbers on we mere humans, scientists often work with numbers using what's called scientific notation. Rather than writing a large or small number as we might normally, we split it into two parts: a mantissa and an exponent. Rather than muddy the waters with an inadequate explanation, I'll give an example.

According to my calculator (and Google), the speed of light in a vacuum is 299,792,458 m/s. When working in scientific notation we write the speed of light as:

2.99792458 ×10⁸ m/s.

Here '2.99792458' is the mantissa and '8' is the exponent. Scientific notation means exactly what is says: take the mantissa, and multiply it by 10^exponent. The multiplier is always 10, so this is the same thing as moving the decimal point exponent places to the left or right.

Using scientific notation helps us write numbers with many non-significant digits in a more compact form (the weight of an electron, for instance, begins with thirty-one 0's and the mass of the Earth ends with twenty 0's), helps us determine the order of magnitude of a number (instead of counting the digits, we can just look at the exponent), and reduces the effect of transcription errors (if we skip, transpose, or mistake one of the digits in the mantissa, the number will at least be in the same order of magnitude as the original).

For more information, you might like to see the Wikipedia articles on scientific notation and order of magnitude and the Wolfram MathWorld articles on scientific notation and order of magnitude.

The Units in Short

To ensure that all students are starting at the same point, the first topic to be covered is the Internation System or Units, or SI. The SI specifies units to measure every conceivable quantity and a range of prefixes to help make these measurements more manageable.

The SI uses seven base units in terms of which all other units can be defined (although two of these, the candela and the metre, are now defined in terms of other base units, they are still know as such for historical reasons):

metre (m)

length

kilogram (kg)

mass

second (s)

time

ampere (A)

electrical current

kelvin (K)

absolute temperature

mole (mol)

quanity of matter

candela (cd)

luminous intensity

The three that we will use most frequently are the metre, the kilogram and the second.

In addition to the seven base units, there are numerous derived units so called because they are derived from the base units. The derived units we'll see include

joule (J)

energy, work

newton (N)

force

In addition to the seven base units and various derived units, the SI defines a number of prefixes to make it easier to deal with large and small numbers. Each prefix specifies an order of magnitude in the same way the exponent in scientific notation (which will be the topic of the next post) does. Thus one millimetre (1 mm) is 1x10-3 m, or 0.001 m. The prefixes we will likely encounter in this course are:

Prefix	Symbol	Exponent	Multiplier
tera	T	12	1 000 000 000 000
giga	G	9	1 000 000 000
mega	M	6	1 000 000
kilo	k	3	1 000
hecto	h	2	100
deca	da	1	10
		0	1
deci	d	-1	0.1
centi	c	-2	0.01
milli	m	-3	0.001
micro	µ	-6	0.000 001
nano	n	-9	0.000 000 001

A Brief History of Physics

Aristotle was perhaps the first in the Western tradition to look at mechanics in any sort of structured way. A philosopher, rather than physicist, Aristotle thought about the way objects interact with each other, particularly their motions.

One of the ideas to come from Aristotle's work is that objects "like" to remain at rest. This seems rather reasonable — put a book on a table and it remains still, push it gently and it will move until you stop. This begs the question, though — what happens when we throw ad object? Our hand stops pushing, but the object continues to move. Likewise when we roll a ball — we release the ball and it continues to move. Aristotle's answer was impetus.

When an object is moved by another (your hand, for example, throwing a ball), it accrues impetus. When the mover stops acting upon the movee, the impetus it accrued whilst being acted upon is used to continue the motion. Under this model, we would expect objects to exhibit straight-line trajectories (see figure one) rather than the parabolic trajectories (see figure two) we see when we throw an object.

Figure 1: A straight-line trajectory of the sort predicted by Aristotle's theory of impetus.

Figure 2: A parabolic trajectory of the sort predicted by classical mechanics.

A second idea of Aristotle's is that heavier objects fall faster than lighter objects. It does, at first glance, seem rather reasonable but it is, like the idea of impetus, quite easily shown incorrect.

The Aristotleans didn't bother to take observations or do experiments to support their beliefs and most of those that came after them were content to trust Aristotle. Thus for more than 100 years, our understanding of mechanics was fundamentally flawed. It is the resolution of this flaw that brings the next major milestone in mechanics: experimentation.

During the 16^th and 17^th centuries, Galileo Galilei and other became amongst the first physicists when they used experimentation to confirm and reject their ideas about the motion of objects. Among Galileo's more famous experiments (though the story is now considered to be untrue) is his dropping balls of varing mass of the Leaning Tower of Pisa by which he showed that, contrary to Aristotle's account, the speed of a falling object is independant of its mass. It is precisely this power — to overturn wrong ideas, even if though they have been believed true for centuries, and to suggest a more complete understanding — that makes experimentation so central to all of the sciences.

This experimental focus was not the last development in the physics that we'll be looking at, though it did help pave the way for it. This next and final (for our purposes) leap was due to Newton — using mathematics to describe physics. After that, classical mechanics was essentially complete, with "only" quite a few decades of improvements and polishing before the introduction of relativity and quantum mechanics. It is physics at this level, the state of the art of classical mechanics circa the mid 19^th century, that we'll be studying in this course.

The Course Outline

The first lecture began with the distribution (in three locations several hundred kilometres apart) of textbooks, questionnaires, course outlines, and a sheet of useful data and formulæ. The skeleton of the course is described below, with more detail to come in subsequent posts.

Week One - Introduction

1. Units
2. Scientific Notation
3. Rounding
4. Vectors

Week Two - Kinematics

1. Definitions of s,v and a
2. Graphs of s,v and a
3. Equations of motion
4. Projectile motion

Week Three - Energy

1. Forms of energy
2. Work and power
3. Kinetic energy
4. Gravitational potential energy
5. Conservation of mechanical energy
6. Elastic potential energy

Week Four - Forces and Newton's Laws

1. Forces
2. Newton's first
3. Newton's second
4. Newton's third
5. Inclined plane
6. Friction

Week Five - Momentum, Circular Motion

1. Conservation of momentum

Week Six - Circular Motion

1. Circular motion
2. Torque
3. Angular momentum

There are readings and several problems from the course textbook (Walding, Rapkins and Rossiter's New Century Senior Physics) for each topic covered.

Notes on a Course in Physics

Over the course of the next six weeks, I will be taking an enabling course in physics. Designed to prepare people without a formal education in physics to study it at an undergraduate level, the course aims to cover enough of a high-school physics syllabus to meet university entry requirements in approximately six weeks.

On this blog, I will post notes, links, and other materials that may be of use to others who wish to begin studying physics.