Contents

Measurements in Physics

1. Some definitions

Physics involves logical reasoning. In Physics, we use numbers to have an idea about quantities.

A book weights 1.20 kilograms and has 20.35 centimeters length and 23.0 centimeters width.

The related digits 1.20, 20.35, and 32.0 are numbers. kilograms and centimeters are called dimensions. Each dimension has a unit. We can use meters, centimeters or kilometers to mesure a length. In science, we use meter (m) for length, kilogram (kg) for mass and second (s) for time; these units come from The International System (SI) standards; wich contains seven base units.
Derived units like velocity are the combination of two or more units from the base units of the SI. It happens that others quantities are expressed in others units, but these others units have their equivalent in the base units of the SI standard.

A velocity is expressed in meter per second (m/s).
A force is expressed in Newtton (N). But the Newton stands for One kilogram multiplyed by One meter divided by One second (1 kg . 1 m / 1s)

2. Measurements

Prefixes in metric units

The average distance between an electron and the nucleus in an atom is 0.000 000 000 05 m
This number is very small! we write it as: 0.5 x 10- 10 m
When we say that the atomic number of the hydrogen is 1; that means the weight of 602 300 000 000 000 000 000 000 = atoms in 1 gram.
This number is very large!. We wrte is as 6.023 x 10+23.
We express then the large and the small numbers, difficult to read and to write in powers of ten. We use some prfixes to symbolise some powers of ten as shoown at right:

 Power Prefix Abbrviation 10- 18 atto a 10- 15 femto f 10- 12 pico p 10- 9 nano n 10- 6 micro μ 10- 3 milli m 10- 2 centi c 10- 1 deci d 10 1 deca da 10 3 kilo k 10 6 mega M 10 9 giga G 10 12 tera T 10 15 peta P 10 18 exa E

Conversions

0.5 x 10- 10 m = 0.05 x 10- 9 m = 0.05 nono-meter.

6.023 x 10+23 = 602,300 x 10+18 = 602,300 exa-meters.

3. How to convert?

Rule:

If We have a unit X and we want to convert it in the unit Y, just multiply X by Y, build the conversion factor by deviding the result by Y expressed in X and simplify by X. Here are some examples:

Example 1:

cm = ? m
cm is provided. m is needed. The steps are:
Multiply cm by m → cm x m
Build the conversion factor : devide the result by m expressed in cm ( that is 100 cm) → cm x m /100 cm
Simplify by cm (the two cm in the numerator and the deminator cancel)→ m /100 = 1 x 10 -2 m

Example 2 :

4.5 kg = ? gram
4.5 kg = (4.5 kg x gram)/ gram = (4.5 kg x gram)/ 10 -2 kg = 4.5 kg x 10 +2 grams.

3. Accuracy and precision

As a human being, measuring a dimension is not free of error. But it's important to minimize errors to reach an accurate result. When we measure, we deal with thrre kind of errors:
- Human errors
- Method (or computational or technique) errors
- Instrumental error
Human errors come from personal bias or carelessness in reading instrument, or recording observations, or mathematical calculations. The solution for this problem is to repeat the measurements in order to be certain that they are close to the true value.
The second source of error is due to a method (technique) we utilize to set a result.
The cause of the instrument error is the instrument itself.
All those three experimental sorts of errors are systematic. An error is random when it is beyond our decisions, that is unknown and unpredictable varaiations or changes in experimental situations such as a temperature fluctuation of the environement.

Example:

Human errors: To read 20 cm while measuring the width of a book we make a mistake and consider 8 (inches!).

Method errors: The well known is the parallax; that is the error related to the position of the experimenter regarding the dimention to measure. To give a good value of the temperature, the best position is that the thermometer is parallel to the face of the observer and the level of the mercury perpendicular to the eye.

Instrument error: We cannot have a good measured value from a damaged equipments and devices.

3.1. Acuuracy

The measured dimension is never perfect. But when those three kind of arrors are minimized, we say that we have a good accuracy. The accuracy is related to "how close we are to the true or ideal or accepted value". We can have a high accuracy without be precise.
The precision is another factor. It refers to the degree of exactness when we make a measurement.

The accuracy of a measurement is expressed by the percentage error:

Accuracy = percentage error =
100 x |AcceptedValue - MeasuredValue|/AcceptedValue (%)

3.2. Precision

The precision is related to the measuring instruments. The more an instrument contains divisions, the more the value becomes precise. We get a precise result among repeated measurements.

Example:

If we measure the length of a book with a meterstick divided into millimeters we get a precise value than if it was marked in centimeters. The precision of a measurement is given by the expression of the Average deviation or the Standard deviation of the obtained results:

Average deviation = Σ|AbsoluteDeviation|i/n [from i=1 to i=n]
= Σ|x - xi|/n [from i=1 to i=n]
Where:
n is the number of the repeated measurements
x is the avarage value of the measurements
x - xi is the absolue deviation of the individual result.

A good estimation depends then on the digit we consider for the measurement.

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