Cosmic Adventure 4.7-8 Positioning & Timing in Classical Physics

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POSITIONING IN CLASSICAL MECHANICS

Cosmic Adventure 4.7


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


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


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.


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.


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


3-Dimensional Setup

The powerful feature of this system is that it is able to extend to infinity in space.

𝑋𝑍

𝑌


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


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


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.


Cosmic Adventure 4.8

TIME IN CLASSICAL PHYSICS


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.


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.


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


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


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.


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


𝑥 = 𝑠

𝑦

𝑧

𝑡

So the classical physics of positioning and timing for a single system is prescribed by these classical equations:


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


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′


OBJECTS ON THE MOVE

To be continued on Cosmic Adventure 4.9

Science

Cosmic Adventure 4.7-8 Positioning & Timing in Classical Physics