Introduction to Physics

Lesson 2: Measurement

George Hrabovsky, MAST

From Last Time

In out last lesson we discussed the problem of motion. We explored how we need to establish a reference point and using a standard unit to measure the distance of an abstract particle from that reference point. In a similar way we use a standard unit to measure the time taken by the motion. The distance traveled is given by the formula,

This is called the displacement of the particle. Here is the initial position and is some later position. Similarly, the time expended by the motion is,

This is called the time interval. Here is the initial position and is some later time. The ratio of these quantities is,

This is called the average speed of the particle. We will here adopt the convention that any time a symbol is within the angle brackets we will consider that the average of the variable.

We now have several options of what to talk about next: coordinate systems, the process of measurement, or to move onward into the subject of kinematics. We have decided to explore the process of measurement.

The Nature of Science

Physics is a science, that is its ideas must always be tested to make sure they is are in accord with nature. When we observe some phenomena that interests us so much that we want to explain it, we begin by gathering information about the phenomenon under study; either we look it up, or we make a measurement. Once we gather this information we look for possible patterns in any available data. If we find such a pattern, we assume it is true and make certain that it makes sense with what we already understand. Then we extend the idea to something we have not yet seen, thus, we make a prediction. This prediction is them tested by experiments. Put another way, science begins and ends with nature.

The Nature of Mathematics and Science

Mathematics is not science. Physics is also not mathematics. In mathematics you must either assume something is true, or prove it to be true before continuing. In physics, if you are uncertain you just go out and measure something. You can, and must, compare all calculations and derivations with reality.

The goal of the mathematician is to make new mathematics. The goal of the physicist is to try to understand nature. The physicist uses mathematics because when a mathematician proves a statement about mathematics the physicist can use all of that mathematical precision to describe physical processes.

If we say that both force and velocity behave like vectors, then we can use all of the mathematical machinery of vector analysis to describe forces and velocities. We now know that we can add forces in exactly the same way that we add vectors. If we measure how a combination of forces ends up, we may be able to deduce what the individual forces must be.

The whole point is the ability to simplify problems by making them more abstract. This is a process of ultimate simplification. The physicist removes all complications imposed by reality to understand the simplest case. Once this is understood, we can begin to put the complications of reality back into system under study piece-by-piece, attempting to understand how each piece changes the system.

For our study of motion we will abstract the system to ignore the size and shape of the moving object and we will begin by thinking of motion only in one direction.

Measuring Time

Last time we discussed time intervals as the difference from the time of measurement to some earlier time. How do we measure time? We use a clock. What is a clock? Hmmm. Think about it a little while. Eventually you will realize that any regular, periodic, phenomena can be a clock. The swing of a pendulum, the tension being released from a spring held mechanically, the vibrations of atoms, laser beams reflecting in mirror systems, and so on.

You can visit the NIST (National Institute for Standards and Technology) site to learn about their atomic clocks at http://tf.nist.gov/cesium/fountain.htm. If you have not visited this web site, do so! It is a gold mine of freely available information absolutely useful for the amateur scientist. We here at MAST have a digital oscilloscope that can measure time intervals as small as a picosecond, seconds. For now we will consider clocks as having fixed time intervals, though we know they do not; Einstein taught us that when a clock moves very fast it will run slower.

Measuring Distance

Last time we discussed displacement as the difference of the position of an object from some earlier position. How do we measure distance? We use a ruler. What is a ruler? Anything of fixed length can be a ruler. Aside from physical objects we can use light as a ruler. We know the speed of light, we shoot a laser beam at a target and measure the time the beam takes to bounce back, and that gives us the distance. The more carefully we measure the time the beam takes, the better the range will be calculated.

Astronomers and the military use a method where two measurements of the angle to a target, say a star, are made a fixed distance apart. If you know the distance between the angle measuring stations and the angles measured, you can use trigonometry to find the distance to the star. This is called the parallax method. In fact, a unit of measurement has been invented just for this method, when a star is seen to move in parallax by one arc second in half a year that distance is called a parsec (parallax-second).

Moving to the very small, we find that things are very mysterious. After a point we can no longer see things with light, and we must use other methods to resolve distances smaller than a wavelength of light. Electrons (charged particles that help to form atoms) can be substituted for light, as they can resolve really small details; this is the idea behind an electron microscope. Atomic force microscopes can measure distances that are even smaller, they sit atop the electric fields of atoms and can measure individual atoms.

Do physical rulers have fixed lengths? In a manner similar to clocks, distances contract as you get closer to the speed of light. Until we get to relativity, we will consider distances as fixed.

When we get very small (on the order of meters or smaller), we find that things get really strange! We can no longer accurately measure both the position of an object and its speed. This is not an imposition of measurement error; it is an imposition of physical law. This means that measurements of time and position have less certainty as we approach the realm of quantum physics (the physics of atoms and smaller objects).

Frames of Reference

By considering both time and distance as fixed, we are in a position to describe the world in which motion takes place. This description of the world is exact within our ability to measure. Of course, every measurement is limited in its accuracy by our perceptions and our ability to construct instruments, thus there are always errors in the description of the world. A description of simultaneous distance and time is called a frame of reference.

Exercises

Do any or all of these that are of interest to you.

1) Make a summary sheet for the concepts discussed above.

2) Make a list of the new terms used and what they mean.

3) In your scientific notebook, write, "Analysis of Experiment #1: Calibration Data." Find a standard to measure a similar passage of time as the one you have chosen. Allow an interval of time to pass on both and compare your measuring device to the standard. Do this many times (at least ten and the more the better). Note the results of each trial. This will form the basis for your calibration. Find a standard measure of distance. Use your distance measuring device to measure the standard and record your results. As with your time measurement, do this many times. Calculate the average difference between your devices and the standards and write these results in your notebook. What does this tell you about the measurements you made in the experiment? Note your thoughts and results.

Created by Mathematica  (July 13, 2007)