Measurement is comparison with a standard. For example when we say the length of a certain object is 3 m we are saying the length of the object is 3 times the length of the standard meter. The standard with which the comparison is made is called a unit of measurement. For example the unit of measurement for length is the meter
There are two systems of units. These are the British System and the SI system. SI is an abbreviation for the French phrase 'Systeme Internationale'. The British system is used in the United States while the SI system is used in most of the rest of the world. For scientific purposes, the SI system is used. In this course, the SI system will be used.
The SI system of units are the units based on standards kept in an SI office in France. These units may be classified into two: Fundamental units and derived units.
These are the minimum set of units from which all the units of physics can be assembled. The standards for SI fundamental units are kept in the SI office in France. Manufacturers of fundamental units should base their units on these standards. The following table is a list of the fundamental units of physics.
| Physical Quantity | Unit | Abbreviation |
|---|---|---|
| Length | meter | m |
| Time | second | s |
| Mass | kilogram | kg |
| Temperature | degree Kelvin | °K |
| Current | Ampere | A |
The units of length (meter), time (second) and mass (kilogram) are the fundamental unit of Mechanics. The unit of temperature (°K) is the fundamental unit of thermodynamics (study of heat). The unit of current (A) is the fundamental unit of electricity and magnetism.
Derived units are units that can be expressed as a combination of fundamental units. For example the unit of speed is a derived unit because it can be expressed as a ratio between the unit of length and the unit of time (m / s). The following table is a list of some derived units of physics.
| Physical Quantity | Unit | Abbreviation |
|---|---|---|
| Speed | meter/second | m/s |
| Acceleration | meter/second2 | m/s2 |
| Volume | meter3 | m3 |
| Density | kilogram/meter3 | kg/m3 |
The units of physics are defined in such a way that they can be comprehended by human senses. For example the kilogram is a weight we can hold in our hand; the second is an interval of time we can comprehend and the meter is about twice of human arm. But in physics, quite often we deal with quantities which are either much much bigger or much much smaller than these units. To deal with such quantities conveniently, names and abbreviations for some powers of ten are defined. For example the "kilo" and "k" are the name and the abbreviation for a 1000. The following table is a list of some commonly used powers of ten.
| Power of Ten | Name | Abbreviation |
|---|---|---|
| 10 3 | kilo | k |
| 10 6 | Mega | M |
| 10 9 | Gega | G |
| 10 -1 | deci | d |
| 10 -2 | centi | c |
| 10 -3 | milli | m |
| 10-6 | micro | µ |
| 10-9 | nano | n |
Significant figures are digits of a report of a measurement that make sense. For example let say the length of a certain object is measured by a ruler whose least count is a cm and the measurement is reported as 4.321 cm. Using this device, it is possible to determine that the length of the object is between 4 cm and 5 cm. The ones digit (4) can be determined exactly. The tenth digit (3) can be determined approximately because we know the length is between 4 cm and 5 cm. Since we are not sure of the tenth digit (3), it is impossible to determine the hundredth digit (2) and the thousandth digit (1). Thus, we say the ones digit (4) and the tenth digit (3) are significant digits while the hundredth digit (2) and the thousandth digit (1) are not significant digits.
Scientifically, significant digits include all accurate digits and one uncertain digit. A report of a measurement should include only significant digits. For example the length of the object in our example should be reported as 4.3 cm. But sometimes we will be forced to include zeros that are not significant digits because zeros are used to hold decimal places. For example let say the length of an object is measured by a device whose least count is 100 cm and its length is found to be between 200 cm and 300 cm. The tens digit can be approximated (let say it is 5) but the ones digit can not be determined. Even though the ones digit can not be determined we have to put zero in its place in order to indicate that the digit 2 is a hundreds digit. The measurement is reported as 250 cm. It is important that we are able to tell whether a zero included in a report of a measurement is significant or not
We can determine whether a zero included in a report of a measurement is significant or not by following the following rules.
Example: How many significant digits are there in the follwing reports of a measurement.
3.1000
Solution: five, because the zeroes are tailing zeroes after the decimal point.
When operating (adding, subtracting, multiplying, dividing) with significant figures, the result can not be more accurate (have greater number of significant digits) than either of the figures being operated.
The result of adding or subtacting significant figures should have the same number of decimal places as the figure with the least number of decimal places. For example when adding 2.13 and 3.4571, the sum should have only two decimal places because one of the figure (2.13) has two decimal places (1 and 3) and the second number (3.4571) has four decimal places (4, 5, 7 and 1). Eventhough the algebraic addition of the figures gives 5.5871, to obtain significant figures this should be rounded to two decimal places and the result should be reported as 5.59.
The result of multiplying or dividing significant numbers should have the same number of significant digits as the figure with the least number of significant digits. For example when multiplying 200 by 38, the result should have only one significant digit because one of the figures (200) has only one significant digit and the other figure (38) has two significant digits. Eventhough direct multiplication gives 7600, to obtain significant figures, this should be rounded to ten thousands decimal place and the result should be reported as 8000.
Expressing a number in standard notation means expressing a number as a product between a number between one (inclusive) and ten and a power of ten. It allows you to express a report of a measurement as a product of a number that consists of significant digits only and a power of ten. The non-significant zeroes are absorbed in the power of ten. For example to express 2400 in standard notation, first we have to divide it by 1000 to get a number between one(inclusive) and ten. This gives 2.4. And then of course we have to multiply by a 1000 or 103 to represent the original number (2400). Thus the standard notation of 2400 is 2.4 x 103. Similarly, to express 0.0540 in standard notation first we multiply it by 100 to change it to a number between one(inclusive) and ten which gives 5.40 and then multiply it by 10 to the power 0f -2. Thus its standard notation is 5.40 x 10 -2. The zero is included in 5.40 because it is significant.
To add or subtract numbers in standard notation, first manupulate the numbers so that all of them have the same pwers of ten. Then, factor out the power of ten and operate. For example to add the numbers 2 x 102 and 3 x 103, first change the power of ten of the first number to 3 by dividing the 2 by 10 and multiplying the power of ten by 10. This gives 0.2 x 103. Then factor out the power of ten to get (0.2 + 3) x 103. And the result is 3.2 x 103.
To multiply or divide numbers in standard notation, multiply (divide) numbers with numbers and powers of ten with powers of ten. For example to multiply 2 x 102 and 3 x 103, multiply the numbers (2 and 3) together and the powers of ten ( 102 and 103 ) together to get 6 x 105.
The following list of laws of exponents may be useful in operating with numbers in standard notation:
x a x b = x a + b
x a ⁄ x b = x a - b
x 0 = 1
x - a = 1 ⁄ x a
( x a ) b = x ab
x a y a = (xy) a
x a ⁄ y a