MeasurementsOne of the things that scientists do is make predictions - predictions based on their hypotheses, laws, and theories. The test of a prediction is whether it works in the "real world" - do the results of experiments match the theoretical prediction? If the results don't match (and these results are confirmed by other competent scientists) then the hypothesis that generated the prediction must be modified or abandoned. Physics is a quantitative science. Physicists deal in numbers - but not just the numbers of the mathematician. This is an important point that is often missed by beginning physicists. Physicist's numbers are often (or could be) measurements, not the pure numbers of the mathematician.

Therefore, physicists measure things. Measurement is very important in physics - physicists are serious about measurement. One of the major contributions of physics to other sciences and society are the many measuring devices and techniques that physics has developed. In "everyday life," we pick up a ruler and measure something without giving it much thought. Physicists think about their measurements, and need to have a much more sophisticated understanding of the measurement process than "normal" people do.

As a physicist, you have to deal with four different types of numbers: Pure "theoretical" numbers Counts

Measurements

Calculated values

Pure Numbers:

Physicists often deal with pure numbers, which are the numbers of the mathematician. Consider the formula for kinetic energy:

In this expression, the fraction "1/2" and the exponent "2" are pure, exact numbers. The exponent isn't approximately 2, it's 2, period. This is not a difficult concept for the beginning physicist, because they generally think (because of their previous training, unfortunately) that all numbers work that way. Big news for you: That is not the case.

Counts:

Experimental physicists often need to count things. For instance, in an experiment on simple harmonic motion, you will generally need to count the vibrations of an object. Now, this is a measurement of sorts, but a measurement that does not (usually) need a measuring instrument. Also, its result is always a whole number.

Usually, we consider counts to be exact values in that there is no measurement uncertainty generally associated with a count. Certainly it is possible to make a mistake in counting, but good technique suggests that counts be checked - two people can count, instead of just one, for instance.

Measurements:

Looking back at the formula for kinetic energy, "m" represents mass. In order to "plug something in for m", the mass of an object has to be measured. It comes as quite a shock to many beginning physicists that no measurement is exact. All measurements have an associated uncertainty, and a good deal of the job of the experimental physicist is determining what that uncertainty is.

Calculated Values:

Looking back at the formula for kinetic energy once more, the symbols "K" and "v" represent still another type of number that the experimental physicist needs to deal with - they are calculated values. To calculate the speed, v, of an object, you need to divide the distance it traveled (a measurement) by the time required (another measurement). In order to calculate the kinetic energy of an object you multiply its mass (a measurement) times its speed (another measurement) to the second (a pure number) power, then divide the product by two (another pure number). If there are uncertainties associated with the distance and time measurements - and there are - then there will be an uncertainty in the speed of the object - but how much uncertainty? By the same token, how does the uncertainty in the measurement of the object's mass and speed affect what we know about its kinetic energy? This sort of analysis is called "propagation of errors", and you need to know something about error propagation in order to make intelligent analyses of experimental data.

Physical QuantitiesA physical quantity is a property that can be measured. The speed of a rocket, the area of a soccer field, and the temperature of a hot cup of coffee are all examples of Physical quantities.

To completely express a physical quantity, it must be quoted with1) A numerical value and 2) A unit

E.g. Distance = 50 m

All physical quantities in Physics can be classified into:

“50” is the numerical value and “m” is the unit. Units are important because the expression “50” is meaningless without it.

(i) Base or fundamental quantities and(ii) Derived quantities.

Base Quantities and UnitsIn the SI system of units, there are seven base quantities and corresponding units.

To make life easier for all, it is agreed that all nations will use the SI (Système International) units as the system of measurement.

Derived Quantities and UnitsThese are physical quantities which are derived from the seven base quantities by mathematical operations such as multiplication, division. Their units are similarly derived as products or quotients of the seven base units.

Measurement Techniques

More than one type of instrument may be used to measure the same physical quantity:

However, no instrument can claim to measure a physical quantity exactly. There will always be some kind of experimental error and a certain degree of uncertainty.

Experimental ErrorsExperimental errors are deviation of measurements from their true values. They are generally divided into two categories, random and systematic.

Random ErrorRandom errors are said to occur when repeated measurements of the same quantity give rise to errors of different magnitude and sign. Random error occurs in an unpredictable manner. It has an

equal probability of being positive or negative in sign. If a measurement is repeated many times and a graph is plotted of the number of occurrences with the values obtained, it may look as follows. There will be a spread of values obtained and the spread occurs around an average value. Such a spread shows the presence of random errors.

Random errors often arise due to natural fluctuations in processes or environmental conditions. These include:

Irregularities in specimens (e.g. the diameter of a wire at various points may be slightly different)

The human reaction time in using stopwatches (e.g. when timing the period of an oscillation, there will be a random error in judging exactly when the oscillation ends)

The number of decay per unit time measured in a radioactive substance Changes in temperature or pressure

Although random errors can never be eliminated, they can be reduced by: Taking a large number of readings and then taking the average, or Repeating measurements and drawing a line of best fit through the plotted points on a suitable

graph.

Systematic Error

Systematic errors are said to occur when repeated measurements of the same quantity under the same conditions give rise to errors of the same magnitude and sign. Systematic errors are consistent errors that cause all measurements to be incorrect by the same amount. They are predictable since they follow some fixed rule or pattern. Repeating the experiment under the same conditions will yield the same errors.

Typical systematic errors include:

zero errors (e.g. using an ammeter with zero reading of -0.2 A will result in all readings taken to be 0.2 A too small),

Incorrect calibration of instruments, Bias of the observer, Incorrect experimental technique (e.g. poor skills resulting in parallax errors that affect all the

readings in the same way).

Unlike random errors, taking a large number of readings and averaging will NOT reduce systematic errors. Fortunately however, systematic errors can be eliminated if the cause of that error is known and corrections are made to the readings. For example,

To correct for zero errors, we subtract the zero reading from all subsequent readings we take.

For instruments that are suspected to be incorrectly calibrated, we compare them with other instruments that are known to be correctly calibrated.

Summarizing the Differences

Dimension:In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.