Units and Measurements

Important Points:

1. Measurement of Length:

a) A meter scale is used for lengths from10-3m to 102m.

b) A Vernier calipers is used for lengths to an accuracy of 10-4m.

c) A screw gauge and a spherometer are used to measure lengths as less as to 10 - 5 m.

d) Large distances such as the distance of a planet or a star from the earth can be measured by parallax method.

1 Parsec = 3.08 x 1016 m

e) To measure a very small size like that of a molecule (10-8 m to 10-10m) electron microscope can be used .Its resolution is about 0.6A0.

     1 Fermi = 1 f = 10-15m

2. Measurement of Mass:

The mass of atoms and molecules are expressed in the unified atomic mass unit (u).

1 Unified Atomic Mass Unit = 1.66 x 10-27kg

3. Measurement of Time:

Cesium clock (or) atomic clock is based on the periodic vibrations of cesium atom. These clocks are very accurate. A cesium atomic clock at National Physical Laboratory (NPL), New Delhi, is being used to maintain the Indian standard of time.

4. Fundamental Quantities:

A physical quantity which is independent of any other physical quantity is called fundamental quantity.

5. Derived Quantities:

Quantities that are derived from the fundamental quantities are called derived quantities.

6. Dimensions:

Dimensions are the powers to which the fundamental units are to be raised to get one unit of the physical quantity.

7. Dimensional Formula:

Dimensional formula is an expression showing the relation between fundamental and derived quantities.

8. Dimensional Constant:

Constants having dimensional formulae are called dimensional constants.

Ex:- Planck’s constant, universal gravitational constant.

9. Dimensionless Quantities:

Quantities having no dimensions are called dimensionless quantities.

Ex: Angle, strain.

10. Numerical value of a physical quantity is inversely proportional to its unit.

11. Principle of Homogeneity:

Quantities having same dimensions can only be added or subtracted or equated.

12. Uses of Dimensional Formulae:

Dimensional formulae can be useda)

a)To check the correctness of the formula or an equation.

b) To convert one system of units into another system.

c) To derive the relations among different physical quantities.

13. Limitations of Dimensional Methods:

1) These cannot be used for trigonometric, exponential and logarithmic functions.

2) These cannot be used to find proportionality constants.

3) If an equation is the sum or difference of two or more quantities, then these methods are not applicable.

4) If any side of the equation contains more than three variables, then these methods are not applicable.

14. Accuracy:

It is the closeness of the measured values to the true value.

15. Precision:

It is the closeness of the measured values with each other.

16. Errors:

The uncertainty in a measurement is called ‘Errors’. Or

It is difference between the measured and the true values of a physical quantity.

17. Types of Errors:

Errors in measurement can be broadly classified as

a) Systematic Errors b) Random Errors

18. Systematic Errors:

a) Instrumental Errors: It arises from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument.

b) Imperfection in experimental technique or procedure.

19. Random Errors:

These errors are due to random and unpredictable fluctuations in experimental conduction, personal errors by the observer.

20. Least Count Errors:

It is the error associated with the resolution of the instrument.

21. Significant Figures:

Significant figures in a measurement are defined as the number of digits that are known reliably plus the uncertain digit.

22. Rules for determining the number of significant figures:

1. All non - zero digits are significant.

E.g. Number of SF in 9864, 9.864, 98.64 is 4.

2. All zeros occurring between two non-zero digits are significant.

E.g. Number of SF in 1.0605, 106.05; 1.0605 is 5.

3. All the zeros to the right of the decimal point but to the left of the first non zero digit are not significant.

E.g. Number of SF in 0.0203 is 3

4. All zeros to the right of the last non zero digit in a number after the decimal point are significant.

E.g. Number of SF in 0.020 is 2.

5. All zeros to the right of the last non zero digit in a number having no decimal point

are not significant

E.g. Number of SF in 2030 is 3.

23. Rounding Off:

The process of omitting the non significant digits and retaining only the desired number of significant digits, incorporating the required modifications to the last significant digit is called 'Rounding Off The Number'.

24. Rules for rounding off Numbers:

1. The preceding digit is raised by 1 if the immediate insignificant digit to be dropped is more than 5.

E.g.: 4928 is rounded off to three significant figures as 4930.

2. The preceding digit is to be left unchanged if the immediate insignificant digit to be dropped is less than 5.

E.g. 4728 is rounded off to two significant figures as 4700.

3. If the immediate insignificant digit to be dropped is 5 then

a) If the preceding digit is even, it is to be unchanged and 5 is dropped

E.g. 4.728 is to be rounded off to two decimal places as 4.72.

b) If the preceding digit is odd, it is to be raised by 1.

E.g. 4.7158 is rounded off to two decimal places as 4.72.