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.

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Physical World

Important Points:

1. Science is exploring, experimenting and predicting from what we see around us.

2. The word Physics comes from a Greek word physis meaning nature.

3. Gravitational Force:
a) The gravitational force is the mutual force of attraction between any two objects by virtue of their masses. It is a universal force.
b) It plays an important role in the formation and evolution of stars, galaxies and galactic clusters.

4. Electromagnetic Force:
a) For a fixed distance, electromagnetic force between protons is 1036 times the gravitational force between them.
b) Electromagnetic force is the base for the structure of atoms and molecules.
c) Gravity is always attractive, while electromagnetic force may be attractive or repulsive.

5. Strong Nuclear Force:
a) The strong nuclear force binds protons and neutrons in a nucleus.
b) This is strongest of all fundamental forces and about 100 times stronger than electromagnetic force.
c) It is charge independent and acts equally between Proton - Proton, Neutron -
Neutron, and Proton -Neutron.
d) Its range is very small (10–15m).
e) It is responsible for the stability of nuclei.

6. Weak Nuclear Force:
a) This force appears only in certain nuclear processes such as the b - decay.
b) These are weaker than the strong nuclear and electromagnetic forces, but stronger than gravitational forces.
c) Their range is very small (10–16 m).

7. The Raman Effect deals with scattering of light by molecules of a medium when they are excited to vibration energy levels.

8. According to Bose-Einstein statistics a gas of molecule below a certain temperature undergoes a phase transition to a state where a large fraction of atoms populate the same lowest energy state.

Very Short Answer Questions

1. What is physics?
A. Physics is the study of basic laws of nature and their manifestation in different natural phenomenon.
2. What is the discovery of C.V. Raman?
A: C.V. Raman discovered Raman Effect. It deals with scattering of light by molecules of a medium when they are excited to vibrational energy levels.

3. What are the fundamental forces in Nature?
A: 1) Gravitational Force
     2) Electromagnetic Force
     3) Strong Nuclear Force and
     4) Weak Nuclear Force

4. Which of the following has Symmetry?
(a) Acceleration due to Gravity (b) Law of Gravitation
A: Law of Gravitation.

5. What is the contribution of S. Chandrasekhar to Physics?
A: Chandrasekhar limit. He worked on structure and evolution of stars.

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Matrices

In mathematics, a matrix (plural matrices) is a rectangular array[1] of numbers, symbols, or expressions, arranged in rows and columns.[2][3] The individual items in a matrix are called its elements or entries. An example of a matrix with 2 rows and 3 columns is

Matrices of the same size can be added or subtracted element by element. But the rule for matrix multiplication is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation. If R is a rotation matrix and v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of a system of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations.

Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[4] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.

A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.

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Mean Value Theorems

In mathematics, the mean value theorem states, roughly: that given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

More precisely, if a function f is continuous on the closed interval [a, b], where a < b, and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that

A special case of this theorem was first described by Parameshvara (1370–1460) from the Kerala school of astronomy and mathematics in his commentaries on Govindasvāmi and Bhaskara II.The mean value theorem in its modern form was later stated by Augustin Louis Cauchy (1789–1857). It is one of the most important results in differential calculus, as well as one of the most important theorems in mathematical analysis, and is useful in proving the fundamental theorem of calculus. The mean value theorem follows from the more specific statement of Rolle's theorem, and can be used to prove the more general statement of Taylor's theorem (with Lagrange form of the remainder term).

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