Measurements¶
Steps to Measure Quantities¶
- Comparison of quantity to be measured with a standard measuring tool.
- Encoding the result of comparison in standard format containing a suitable precise number along with appropriate unit.
- Basic quantities could not be expressed in term of other quantities
- Derived quantities are based on the other physical quantities
- Measurement of basic quantities requires the choice of standard. Ideal standard should be accessible & invariable.
Prior 1960, three measuring systems were in practice- CGS System (centimeter, gram, second)
- FPS System (foot, pound, second), or British Engineering system.
- MKS System (meter, kilogram, second)
- After 1960, internationally accepted standard is SI system. In system international (SI), there are three types of units.
Info
Ideal standard measuring tool should be fool proof via accessiblity & invariability.
Supplementary Units¶
| Physics Quantity | SI Unit | Symbol |
|---|---|---|
| Plane angle | Radian | \(rad\) |
| Solid Angle | Steradian | \(sr\) |
Basic Mechanical Units¶
| Name | SI Unit (MKS) | CGS | US Common |
|---|---|---|---|
| Length (\(L\)) | meter (\(m\)) | centimeter (\(cm\)) | foot (\(ft\)) |
| Time (\(T\)) | second (\(s\)) | second (\(s\)) | second (\(s\)) |
| Mass (\(M\)) | kilogram (\(kg\)) | gram (\(gm\)) | slug |
| Velocity (\(L/T\)) | \(m/s\) | \(cm/s\) | \(ft/s\) |
| Acceleration (\(L/T^2\)) | \(m/s^2\) | \(cm/s^2\) | \(ft/s^2\) |
| Force (\(ML/T^2\)) | \(kg\;m/s^2=N\) | \(gm\;cm/s^2=dyne\) | \(slug\;ft/s^2=pound(lb)\) |
| Work (\(ML^2/T^2\)) | \(N\;m=J\) | \(dyne\;cm=erg\) | \(lb\;ft=ft\;lb\) |
| Energy (\(ML^2/T^2\)) | \(J\) | \(erg\) | \(ft\;lb\) |
| Power (\(ML^2/T^3\)) | \(J/s=watt(W)\) | \(erg/s\) | \(ft\;lb/s\) |
Note
Many other derived units are also used for other derived quantities.
Scientific Notation¶
The standard to express the number in term of power of ten is called Scientific Notation.
Some standard prefixes for power of ten with their symbols are presented below:
| Factor | Prefix | Symbol | Factor | Prefix | Symbol |
|---|---|---|---|---|---|
| \(10^{-18}\) | atto | \(a\) | \(10^{1}\) | deca | \(da\) |
| \(10^{-15}\) | femto | \(f\) | \(10^{3}\) | kilo | \(k\) |
| \(10^{-12}\) | pico | \(p\) | \(10^{6}\) | mega | \(M\) |
| \(10^{-9}\) | nano | \(n\) | \(10^{9}\) | giga | \(G\) |
| \(10^{-6}\) | micro | \(u\) | \(10^{12}\) | tera | \(T\) |
| \(10^{-3}\) | milli | \(m\) | \(10^{15}\) | peta | \(P\) |
| \(10^{-2}\) | centi | \(c\) | \(10^{18}\) | exa | \(E\) |
| \(10^{-1}\) | deci | \(d\) |
Do you Know?
There should be one non-zero digit to left of decimal in scientific notation.
Error¶
The difference between standard and experimental values is called error.
Classification of Errors¶
- Assignable Errors (Systematic Errors)
- Unassignable Errors (Random Errors)
Assignable Errors¶
The errors to which we can assign a cause usually they follow same trend of variations so a cause can be assigned to them eg. Errors occurring due to the LC of the measuring instrument.
Remedy¶
Such errors can be controlled experimentally while carefully monitoring the measurement process.
Unassignable Error¶
To which we can't assign a cause because they do not follow a trend. So these errors are due to unknown causes.
Remedy¶
These errors can't be controlled experimentally, but a statistical tool of taking average of several value is employed to minimize them.
Significant figures¶
Uncertainty in the measurements leads us to establish some standard to write the numerical value of a measurement, termed as significant figure.
"In any measurement, all the accurately known digits and the first doubtful digit are named as significant figures."
Rules of significant figures¶
In order to determine significant figures in a number we must follow the following rules:
* All the non-zero digits are significant figures.
For Example:
3,456 has four significant figures.
12.3456 has six significant figures.
* Zeros between non-zero digits are significant.
For Example:
2306 has four significant figures.
20,0894 has six significant figures.
* Zeros locating the position of decimal in numbers o not significant.
For Example:
In the 0.2224 and 0.000034, zeros are not significant
* Final zeros to the right of the decimal point are significant.
For Example:
3.0000 has four significant zeros.
1002.00 has four significant zeros.
* Zeros that locate decimal point in numbers greater than one are not significant.
For Example:
3000 has only one significant figure.
120000 has two significant figures.
Do you know?
Greater the significant figures in measurement the more accurate it is.
Algebra with Significant Figures¶
Division and Multiplication¶
- \(\frac{4.54\times 2.324}{1.3365} = 7.89447063 =\) an answer having maximum of three digits can he retained here \(= 7.89\).
- \(4.3458\times 2.7 = 11.73366 =\) an answer having maximum of two digits can he retained here \(= 12\)
- The factor having the smallest number of significant figures is called least accurate factor, and product and quotient cannot have number of significant figures more than that in the least accurate factor
Addition and Subtraction¶
Search for the least number of decimal places, counting the number of significant figures is not required. For example:
- \(4.345+ 23.51= 27.855 =\) an answer having maximum of two decimal places \(= 27.86\)
- \(101.2401-1.0=100.2401=\) an answer having maximum of one decimal place \(=100.2\)
- \(101.2401-1=100.2401=\) an answer having no decimal place \(=100\)
Rounding Off Data¶
Rule # 1:¶
If the digit to be dropped is greater than 5, then add 1 to the last digit to b, retained and drop all digits farther to the right.
For example:
- \(3.677\) is rounded off to \(3.68\) if we need three significant figures in measurement.
- \(3.677\) is rounded off to \(3.7\) if we need two significant figures in measurement.
Rule # 2:¶
If the digit to be dropped is less than 5, then simply drop it without adding any digit number to the last digit.
For example:
- \(6.632\) is rounded off to \(6,63\) if we need three significant figures in measurement.
- \(6.632\) is rounded off to \(6.6\) if we need two significant figures in measurement.
Rule # 3:¶
If the digit to be dropped is exactly 5 then:
Case 1¶
If the digit to be retained is even, then just drop the "5".
For example:
- \(6.65\) is rounded off to \(6.6\) if we need two significant figures in measurement.
- \(3.4665\) is rounded off to \(6.466\) if we need four significant figures in measurement.
Case 1¶
If the digit to be retained is odd, then add 1 to it.
For example:
- \(6.35\) is rounded off to \(6.4\) if we need two significant figures in measurement.
- \(3.4675\) is rounded off to \(6.468\) if we need four significant figures in measurement:
Note
Zero is an even number So, \(3.05\) is rounded off to \(3.0\) if we need two significant figures in measurement.
PRECISION & ACCURACY¶
Precision¶
The precision of a measurement is determined by the least count of the instrument or device.
Smaller the least count of instrument, more precise is the measurement.
Accuracy¶
The accuracy of a measurement depends upon the fractional or percentage uncertainty in the measurement.
Smaller the percentage error, more accurate is the measurement. Maximum absolute uncertainty is equal to one least count of the measuring instrument.
Assessment of Total Uncertainty in the Final Result¶
Total uncertainty in the final result can be determined by the following rules:
- In case of addition and subtraction absolute uncertainties are to be added.
- In case of multiplication and division fractional or percentage errors are to be added.
- For power factor the power is multiplied with the percentage uncertainty.
For example: In the equation \(u= kx^a\), \(a \times \text{percentage error in} x \text{, where} k \text{is constant of proportionality}\).
Uncertainty in the average value of many measurements of the same quantity is the mean of absolute differences of measurements form the average value. - For time measurement, uncertainty is obtained by dividing the least count of the timing device with the number of vibrations.
Do you know?
Counting more number of vibrations can reduce the uncertainty in the timing experiments.
Dimensions¶
Each basic measurable physical quantity is represented in term of base quantities by a specific symbol for them written with in square brackets is called dimension.
The dimensions are helpful in:
- Deriving a possible formula
- Checking the homogeneity of a physical equation or formula.
Limitations of Dimensional Analysis¶
- Dimension analysis has no information on dimensionless constants.
- If a quantity is dependent on trigonometric or exponential function, this method cannot be used.
- In some cases, it is difficult to guess the factors while deriving the relation connecting two or more physical quantities.
- This method cannot be used in an equation containing two or more variables with same dimensions.