Upload
stephen-kwong
View
272
Download
0
Embed Size (px)
Citation preview
© ABCC Australia 2015 new-physics.com
Bodily Orientation
When we wish to tell other people about the location of an object, we would say that the object is on our left, right or front to show the relationship of its position to our bodies.
In doing so, we are using our bodies as a set of reference frames to which the object is referred.
Right
Over
Front
Left
Under
Back
© ABCC Australia 2015 new-physics.com
Orientation System
Anyone would know that the terms left, right, front, back, up, and down actually embraced all the direction of space around us.
Left and right are in line with each other and form a pair, so are front and back, up and down. They form three pairs of complementary directions, each pair orthogonal or perpendicular to both of the others.
Right
Over
Front
© ABCC Australia 2015 new-physics.com
Extends to all space
These axes can be extended to any lengths thus covering all space around us. We can point to a close-by house or a distant galaxy in the same manner without the fear of incurring ambiguity.
© ABCC Australia 2015 new-physics.com
Cartesian coordinate system
This set of rectangular coordinates, or bodily coordinates has been used intuitively and unknowingly by people in ancient times, or even in pre-historic times, the same way we use it in our daily life today. But the first person who consolidated the idea into a definable system was the great French philosopher René Descartes (1596 –1650). So the setup is often referred to as the Cartesian coordinate system or simply the Cartesian coordinates.
© ABCC Australia 2015 new-physics.com
3-Dimensional Setup
There are many other kinds of coordinate systems, but the three dimensional Cartesian coordinate system has always been and is still the most popular system used in physics.
𝑋𝑍
𝑌
© ABCC Australia 2015 new-physics.com
3-Dimensional Setup
The powerful feature of this system is that it is able to extend to infinity in space.
𝑋𝑍
𝑌
© ABCC Australia 2015 new-physics.com
But what we are dealing with now is rectilinear motion, so a one dimensional system is good enough to do the job. Sometimes we add the y-axis just to indicate that it is actually a three dimensional coordinate system.
𝑋
𝑌
© ABCC Australia 2015 new-physics.com
Two Dimensional Coordinate System
By dealing with only one dimensions, life is much easier. At most we can add the y-axis to indicate that it is a coordinate system for clarity, so we are using a one or two dimensional system as deemed appropriate from now on.
𝑥0
© ABCC Australia 2015 new-physics.com
0
P
The simplest system
𝑥
So here is our classical reference system in its simplest form. It is just our common sense of locating an object. The object is now at P with coordinate 𝑥. That is it is 𝑥units of length away from the origin O.
© ABCC Australia 2015 new-physics.com
Jury: Besides space, can the classical system also tell how the observers recognize time?
Newton: That is for sure. Classical physics is a complete system that covers both space and time. We do have various devices to measure time. The most popular ones are the water clocks and the sand clocks.
© ABCC Australia 2015 new-physics.com
Your gadgets are not that suitable for our purposes. They are inaccurate and hard to calibre. Here is a clock which is a better device to illustrate our discussions. It has a good mechanical system to make it an excellent piece for timing.
© ABCC Australia 2015 new-physics.com
The Observed Object
The best object to be observed in the study of Relativity is the clock. It offers an addition advantage over any other objects.
Besides being able to be moved from place to place, it can also register the time at the corresponding position. To make things simpler, we use one hand only, that is, we only use clocks that register seconds. So we have now a 12 second clock.
Seconds
12 second clock
Ordinary clock
© ABCC Australia 2015 new-physics.com
0
P
Galileo: That is very nice. It is surely more effective. Thanks Angela!
Time in the classical coordinate system behaves the same as its position. Time at one corner on Earth is the same as time at any other corner.
𝑥
© ABCC Australia 2015 new-physics.com
Synchronized Clocks
But to compare time, we need to first synchronize the clocks.
The best way to synchronize clocks is by setting their hands to exactly the same configurations at the same location.
After this, we can put them in different frames of reference and use them to measure events.
© ABCC Australia 2015 new-physics.com
𝑠
0
In the linear system as we have, the clocks all run at the same rate, thus telling the same time at any instant. So here is the equation which represents time in any location:
𝑡 = 𝑡′
𝑥
© ABCC Australia 2015 new-physics.com
𝑥 = 𝑠
𝑦
𝑧
𝑡
So the classical physics of positioning and timing for a single system is prescribed by these classical equations:
© ABCC Australia 2015 new-physics.com
𝑠
0
𝑥
𝑥’
0′
Two Systems But if there are two systems referring to the same object, then:
𝑥′ = 𝑥 − 𝑠
𝑥 = 𝑥′ + 𝑠
𝑃
© ABCC Australia 2015 new-physics.com
So these are the two sets of equations treasured by classical physics:
System x:
𝑥′ = 𝑥 − 𝑠
𝑦′ = 𝑦
𝑧′ = 𝑧
𝑡′ = 𝑡
System x’:
𝑥 = 𝑥′ + 𝑠
𝑦 = 𝑦′
𝑧 = 𝑧′
𝑡 = 𝑡′