Notes for Physics Chapter 1

CHAPTER 1

MEASUREMENT

Upto the beginning of nineteenth century there was a great increase in the volume of scientific knowledge. Therefore, it was considered necessary to divide the study of nature into two branches, biological sciences which deal with living things and physical sciences which concern with non-living things. Physics, Chemistry, Astronomy and Geology etc are the well known physical sciences while Botany and Zoology are biological sciences. Though all theses sciences are quite useful in their respective field but physics plays the maximum role in the developments of present scientific age.

Physics is an experimental science and the scientific method which emphasized the need of accurate measurement of various measurable features of different phenomena.

1.1 INTRODUCTION TO PHYSICS.

DEFINITION OF PHYSICS:-

Physics is the branch of science which deals with the study of matter, energy and the relationship between them. It explains the nature phenomenon with the help of fundamentals law and principles.

MAIN FRONT LINE OF FUNDAMENTAL SCIENCE:-

There are three main frontiers of fundamental science.

1. The World of extremely large, that is universe, Radio Telescopes now gather information form the far side of the universe about the radio waves and the fire light of the big bang.
2. The world of the extremely small, that of particles such as electrons, protons neutrons, mesons and others.
3. The world of complex matter, it is a world of middle sized things form molecule to Earth

BRANCHES OF PHYSICS

Main branches of Physics are

1. NUCLEAR PHYSICS:-

It is that branch of Physics which deals with atomic nuclei. Nuclear physic is related to neutrons and protons which form the nuclei of atoms (called nucleons), their structure, energy state, reaction between nuclei and radioactivity, for example, nuclear fission, nuclear fusion and nuclear forces.

2. PARTICLE PHYSICS:-

                        It is that important branch of physics which is concerned with the ultimate particles of which the matter is composed Particle physics is related with understanding the properties and behavior of elementary particles.

3. RELATIVISTIC MECHANICS:-

                        It is that branch of physics which deals with velocities approaching that of light. Relativistic mechanics is concerned with the elementary particles related to the special theory of relativity and describe the motion of particles having velocities close to the speed of light (2×10⁸ ms^-1).

4. SOLID STATE PHYSICS

It is that branch of physics which is concerned with the structure and properties of solids, and the phenomenon associated with solids.

ROLE OF PHYSICS IN TECHNOLOGY

Physics has played an important role in the development of technology and engineering science and technology have created a revaluation in the outlook of mankind. All the means of communications and information media have brought all the parts of world in close contact with one another. Events in one part of the world immediately reverberate or spread round the globe.

Now a days, we are living in the age of information technology. The computer network are products of chips developed form the basic ideas of physics. The chips are made of silicon, silcon is obtained form sand. Thus, it helps us to make a computer units.

1.2 PHYSICS QUANTITIES:-

DEFINITION:-

All those quantities in terms of which laws of physics can be described and whose measurement is necessary to understand any problem, are called physical quantities.

TYPES OF PHYSICAL QUANTITIES:-

Physical Quantities have been divides into two categories

1. Base Quantities.
2. Derived Quantities.
3. BASE QUANTITIES:-

Base Quantities are those quantities which can not be defined in terms of other physical quantities. Length, mass and time are typical examples of base quantities.

2. DERIVED QUANTITIES:-

                        Derived quantities are those which are expressed in terms of other physical quantities. Velocity, acceleration, force and momentum are the examples of derived quantities.

MEASUREMENT OF BASE QUANTITY:-

The measurement of base quantity is based on tow steps.

1. Select6ion of a Standard
2. The Establishment of a procedure for comparing the quantity to be measured with the standard so that number and a unit are found as the measure of that quantity.

An Ideal Standard:-

It has two main characteristics

1. It is accessible
2. It is invariable

These tow requirements are often against each other and a compromise has to be mad between them.

1.3 INTERNATIONAL SYSTEM OF UNITS:-

In 1960, an international committee framed a system of units called the system international or international system of units. The units included in this system are known as SI units. The international system of units (SI) is built up form three kinds of units: base units, supplementary units and derived units.

1. Base Units
2. Supplementary Units
3. Derived Units.
4. Base Units:-

In SI there are seven base units of various physical quantities which units used for base quantities are called base units. They are seven as

1. Length ii. Mass iii. Time         iv. Temperature        v. Electric Current
2. Luminous Light (of intensity of light) vii. Amount of Substance (in terms of particles, moles).

The names of base units for these physical quantities are given in the table below:-

2. SUPPLEMENTARY UNITS:-

The General conference on weights and measures has not yet get classified certain units of the St under either base units or derived units these SI units are called supplementary units.

For the time being this class contains only tow units of purely expressed as

1. Plane angle
2. Solid angle

iii.       RADIAN

It is the unit of plane angle.

DEFINITION:-

Radian is defined as the plane angle between two radii of a circle which cut off on the circumference an arc, equal in length to the radius i.e Arc AB

AB= ras shown in the fig 1.1 (a)

Now <AOB = 1 Radian.

(ii)                   STERADIAN:-

It is the unit of solid angle.

DEFINITION:-

Steradian is defined as the solid angle (three dimensional angle)

Subtended at the centre of

a sphere by an area of its surface

equal to the square of radius of the

sphere as shown in fig 1.1 (b) .

3. DERIVED UNITS

DEFINITION  

SI units of all other physical quantities derived form the base and supplementary units are called derived units.

For example, the units of velocity, acceleration, force, work and momentum etc are the derived units because the units of these quantities are the combination of two or more base units.

Some of the derived units are given in the following table 1.3.

EXAMPLES OF DERIVED UNITS

(i) SPEED:-

It is defined as the distance covered in unit time in S.I units, the unit of distance is meter (m) and unit of time is second (s)

Speed =                        =

= m/s = ms^-1

Thus SI unit of speed = ms^-1

(ii) ACCELERATION:-

                        Similarly

Acceleration =

=

= length / (Time)² = m/s² = ms^-2

Thus the SI unit of acceleration = ms^-2

1.4  SCIENTIFIC NOTATION

Numbers are expressed in standard form called scientific notation which use power of ten.

The internationally accepted practice is that there should be only one non-zero digit left of decimal for example

The number 134.7 can be written as

134.7 = 1.347 x 10²

  Similarly 0.0023 can be written as 0.0023 = 2.3 x 10^-3

Solution :-

0023 =                         =                                  = 2.3 x 10-3

USE FOR PREFIX:-

                        Prefixes are used to express these large or small numbers as multiple of ten. For example

1 light year = 946 x 10¹³m              = 9.46 x 10¹⁵m

(946 x 10¹³ x 100 = 1.2 x 10^-15)

100

Similarly for smaller number

Radius of proton = 12 x 10^-16m =1.2 x 10^-15m

(12 x 10^-6 x 10 = 1.2 x 10^-15)

10

1.5  ERRORS AND UNCERTAINTIES:-

Error: –           The Difference between measured all physical measurements are uncertain or imprecise to some limit. Value and Experimental Value is called an error.

There are tow major types of errors which are as follows:

1. Random Error
2. Systematic error.
3. RANDOM ERROR

Random Error is said to take place when repeated measurements of the quantity, give different values under the same conditions. Ti sis due to some unknown reasons.

REDUCTION OF RANDOM ERROR.

The effect of Random Errors can be reduced by taking several readings of same quantity and then taking their mean (or average) value.

2. SYSTEMATIC ERROR

The Systematic errors occur when all the measurements of a particular quantity are affected equally. These give consistent difference in the readings.

OCCURRENCE OF SYSTEMATIC ERRORS

The systematic error can occur due to

(i)                    Zero Error in measuring instruments.

(ii)                   Poor Calibration of instruments or incorrect marking on the measuring instruments.

REDUCTION OF SYSTEMATIC ERROR:-

Systematic error can be reduced by comparing the instruments with another instrument. Which is known to be more accurate.

Uncertainty

The uncertainty is also usually described as an error in an instrument. It can take place due to

1. Inadequacy or Limitation of an instrument
2. Natural Variations of the object being measured.

iii.                   Natural defect of person’s senses

GENERAL RULES

(i)                    All non zero digits (1,2,3,4,5,6,7,8,9) are significant figures.

(ii)                   A zero between two significant figures is itself significant.

(iii)                 Zero to the left of significant figures are not significant. For examples, none of the zeros in 0.00467 or 02.59 is significant.

(iv)                  In decimal fractions, zeros to the right of a significant figure are significant. For example, all the zeros in 3.570 or 7.400 are significant.

(v)                   If the number is an integers i.e a whole number such as 8000 kg, the number of significant zeros is found by the accuracy of the measuring instruments.

If the least count of a measuring scale is 1 kg, then there are four significant figures written in scientific notation as 8.000×10³ kg, if the least count of the scale is 10 kg, then the number of significant figures will be 3 written in scientific notation as 8.00 x 10³ kg and so on.

When a measurement is recorded in scientific notation or standard form, the figures other then the powers of 10 are significant for example a measurement recorded as 8.70 x 10⁴ kg has three significant figures (8,7,0).

MULTIPLICATION AND DIVISION OF NUMBERS:-

In multiplying or dividing numbers, keep a number of significant figures in the product or quotient not more than that contained in the least accurate factor

5.348 x 10^-2 x 3.64 x 10⁴  = 1.45768982 x 10³

1.336

(vi)                  ADDITION OR SUBTRACTION OF NUMBERS:-

In adding and subtracting numbers, the number of decimal places retained in the answer should be equal to the smallest number of decimal places in any of the quantities being added or subtracted.

(i)        72.1                                        (ii)       2.7543

3.42                                                    4.10

0.003                                                  1.273

75.523                                                8.1273

Correct Answer (i) 75.5 m              (ii) 8.13

                        SUBTRACTION

(i)        88.9                                        (ii)       50.5

–   44.32                                                  – 3.2

44.58                                                  47.3

Correct Answer (i) 44.6       (ii) 47 Ans

RULES FOR ROUNDING OF NUMBERS:-

• If the digit to be dropped is greater then 5, the digit to be retained is increased by one. e.g 15.6 is rounded off as 16.
• If the digit to be dropped is greater then 5, the digit to be retained is increased by one e.g 15.6 is rounded off as 16.
• If the digit to be dropped is 5, and if any digit following it is not zero, the previous digit which is to be retained is increased by one e.g 15.51 is rounded off as 16
• If the digit to be dropped is 5, the rounding off date the dropping of last digit and changing the remaining digits according to values is called rounding off data rules.

PRECISION AND ACCURACY

PRECISION:-

                        The closeness of measured value when the same experiment is repeated many times is called precision.

ACCURACY:-

The closeness of measured value to the experimental value is called accuracy.

EXPLANATION:-

The precision of a measurement is determined by the instrument or device being used. The accuracy of a measurement depends upon the fractional or percentage uncertainty in that measurement.

Example:-     When the length of an object is recorded as 25.5 cm by using a meter rod having smallest division in millimeter, it is the difference of tow readings of the initial and final positions. The uncertainty in the single reading as discussed in the previous example is taken as ± 0.05 cm which is now doubled (due to initial and final readings) is called “absolute uncertainty. Thus Absolute uncertainty = ± 0.5 ± 0.5 = ± 0.1 cm.

The absolute certainty is equal to least count of the measuring instruments i.e meter rod. This is also called precision.

First Case:-

Precision or absolute uncertainty ( least count) = ± 0.1 cm.

As the length of the object recorded by a meter rod having least count 0.1 cm is 25.5 cm. then Fractional uncertainty = 0.1 cm =0.004

25.5 cm

Percentage uncertainty = 0.1    x    100     =          0.4       =0.4%

25.5         100                 100

Second case:-

An other measurement of length is taken by vernier caliper with least count as 0.01 cm is recorded as 0.45 cm it has.

Precision or absolute uncertainty (least count) = ± 0.01 cm

Fractional uncertainty = 0.01 cm

0.45 cm

Percentage uncertainty = 0.01   x   100

0.45         100

= 2/100 = 2.0%

This shows that the reading 25.5 cm taken by meter rule is less precise but is more accurate. In fact it si the relative measurement, which is important. The smaller a physical quantity, the more precise instrument should be used. Thus, the measurement 0.45 cm demands that a more precise instrument such as micrometer screw gauge with least count 0.001 cm should be used.

 EXAMPLE 1.1

The length, breadth and thickness of a sheet are 3.23 m, 2.105 m and 1.05 cm respectively, calculate the volume of the sheet correct upto the appropriate significant digits.

SOLUTION:-

                        DATA:-

Length of Sheet = / = 3.233 m

Breadth of Sheet = b = 2.105 m

Thickness of sheet = h = 1.05 cm = 1.05 x 10^-2 m

TO FIND

Volume of the sheet = V = ?

FORMULA:-

V = / x b x h

CALCULATIONS:-

Using the formula for volume

V = / x b x h

and                  putting the values, we get

V = 3.233 x 2.105 x 1.05 x 10^-2 m³

= 7.14573825 m³

As the factor 1.05 cm has the minimum number of significant figures equal to three.

Therefore, volume is recorded upto 3 significant figures,

Hence, Ans.

Example 1.2:-

The mass of a metal box measured by a lever balance is 2.2 kg. two silver coins of masses 10.01g and 10.01g measured by a beam balance are added to it. What is now the total mass of the box correct upto the appropriate precision.

SOLUTION:

                        DATA:-

Mass of a metal box = m₁ = 2.2 kg

Mass of firs silver coin = m₂ = 10.01g = 0.01001 kg

Mass of second silver coin = m₃ = 10.02g = 0.01002 kg

TO FIND

Total mass of the box = M = ?

FORMULA:-

M = m₁ + m₂ = m₃

CALCULATIONS:-

Total mass of the box is found by the addition of silver masses to the mass of box. Thus,

M = m₁ + m₂ = m₃

Putting the value we get

M = 2.2 + 0.01001 + 0.01002

Or                    M = 2.2003 kg

Since least precise is 2.2 kg, having one decimal place, hence total mass should be to one decimal place which is the appropriate precision. Thus,

Ans:

EXAMPLE 1.3

The diameter and length of a metal cylinder measured with the help of vernier calipers of least count 0.01 cm are 1.22 cm and 5.35 cm calculate the volume V of ht ecylinder and uncertainty in it.

SOLUTION:

DATA:-

Least count of vernier calipers = 0.01 cm

Diameter of metal cylinder = d = 1.22 cm

Length of metal cylinder = / = 5.35 cm

TO FIND:-

Volume of the cylinder = V = ?

Uncertainty in the volume = ?

FORMULA:-

V = λd2I

4

CALCULATIONS:-

Absolute uncertainty in length = 0.01 cm

PHYSICAL QUANTITY:-

Property of mater related to its measurement is called

1) Scalars                   2) Vectors

Scalars:-

                        A physical quantity which is completely described by magnitude with proper units is called scalar quantity.

OR

Scalar is physical quantity which has magnitude (Numerical Value) but no direction.

Examples:-   1) Mass                      2) Distance                3) Speed                     4) Energy

5) Work                     6) Valume                  7) Temperature         8) Time

9) Electric Charge    10) Area                     11) Atmospheric Pressure

12) Value of money 13) Potential              14) Wave Length

15) Density                16) Length                 17) Electric Flux

Scalars are added, subtracted, multiplied and divided by ordinary rules of arithmetic.

Vectors:-

A Physical Quantity which is completely described by magnitude with proper units and direction is called Vector quantity.

OR

Vector is a physical quantity which has both magnitude and direction.

Examples:-   1) Displacement       2) Velocity                3) Acceleration

4) Force                     5) Weight                   6) Momentum

7) Torque                   8) Ocean Currents    9) Electric Intensity

10) Angular Velocity.          11) Angular Acceleration.

Vectors cannot be added, subtracted, multiplied and divided by ordinary arithmetic rules but we use a new set of mathematical rules called Vector algebra. Which includes vector addition, vector subtraction and vector multiplication etc. BUT

Remember:- When vectors are parallel (in same direction) and anti-parallel (in opposite direction), then they can be added and subtracted by simple arithmetical rules.

Physical Quantities that have both numerical and directional properties are represented by vectors.

Vector Representation:-

Symbolic Representation:-

A vector can be represented by a:-

(i) Bold face letter    (e.g) A                        Or

(ii) Letter with an arrow head above or below it.    (eg)   A       or     A

If we wish to refer only to the magnitude of a vector we use light face type such as a.

Graphical Representation:-

A vector can be drawn graphically be a straight line with its length representing magnitude of vector and Arrow head indicating the direction of vector.

Two things are required to draw a vector graphically.

1) Suitable Scale

2) Direction indicator:-       Direction indicator are two mutually perpendicular lines

Magnitude of Vector:-

Magnitude of Vector is known as Modulus. It can be represented by placing symbol of vector in within two parallel lines (e.g)  A

RECTANGULAR CO-ORDINATE SYSTEM:-

A set of co-ordinate axis which is used to locate the position of an object (vector) called Rectangular co-ordinate system.

Co-ordinate Axis:- Two lines at right angle to each other are called C-ordinate axis.

Origin:-         Point of intersection of two lines is called Origin.

X-axis:-         Horizontal line is called X-axis. It is also called Horizontal axis. With positive direction to the right.

Y-Axis:-        Vertical line is called Y-Axis. It is also called Vertical axis with positives direction upward.

Z-Axis:-         Third axis which is located in space is called Z-axis. It is right angle to both X-axis and Y-axis.

Two Dimensional Co-ordinate System:-           It shows direction of vector in a plane (e.g) XY plane.

Three Dimensional Co-ordinate system:-         It shows direction of a vector in XYZ plane. This system of Coordinate axes is called as Cartesian or rectangular coordinate system.

Trigonometric Ratios:-

On this right triangle

(i) Sin θ           =         Prep

HYP

(ii) Cos θ        =          Base

HYP

(ii) Tan  θ       =          Prep

Base

KINDS OF VECTORS

Free Vector:            Vectors which do not have fixed position in space so they are usually drawn with respect to geographical direction.

Unit Vector:-           A vector whose magnitude is unity (i.e) 1 is called a unit vector. It is denoted by placing a cap on the symbol of vector (e.g) A is unit vector of A.

Purpose of Unit vector:-   Unit vector is used to represent direction of a vecto.

In three dimensional system:-

• i represents vector along X-axis.
• j represents vector along Y-axis.
• k represents vector along Z-axis.

Two of more frequently used unit vectors are vector r which represents the direction of the vector r and the vector n which represents the direction of a normal drawn on a specified surface.

Magnitude of unit vector A can be determined by flowing formula:-

A = A

A

Equal Vectors:-      Tow vectors having:-

(i) Same Magnitude             (ii) Same Direction   regardless of the position of their initial points are called Equal vectors.

OR

Parallel vectors of equal magnitude are called Equal vectors.

Example:-  

Negative Vectors:- A vector whose magnitude is same as the original vector direction is opposite.

OR

Anti-parallel vectors of equal magnitude are called Negative vectors

Example:-

Null Vector:-

A vector of Zero magnitude and undefined direction is called Null or Zero vector. It is an imaginary vector and cannot be represented along any axis.

Sum of a vetor and its negative vector is a null vector

A + (- A) = 0

Positive Vector:-

A vector which describes the position (distance and direction) of a point form the origin of the co-ordinate system is called the position vector. It is denoted by r.

MULTIPLICATION OF A VECTOR BY A NUMBER ( BY A SCALAR):-

1^st Case:-        when a vector is multiplied by a +ve number (n) then the magnitude of the resultant vector becomes n-times the magnitude of original vector.

But there is no change in direction of resultant vector.

Example:-      A = 2cm (East)  A = 2cm

If         A is multiplied by 3, then

3A = 3 x 2 = 6 cm (East)                        3A = 6cm

2^nd Case:-      when a vector is multiplied by a – ve number (-n) then the magnitude of vector becomes n-times and direction change by 180° (i.e.) becomes opposite.

Example:-      A = 2cm (East)  A = 2cm

If         A is multiplied by -3, then

-3A = -3 x 2 = 6 cm (East)                     -3A = 6cm

ADDITION OF VECTORS BY GRAPHICAL METHODS

Resultant Vector:- The resultant of two or more similar vectors is a single vector which has the same effect as the combined effect of all the vectors to be added. There are three methods of vector addition.

1) Parallelogram Rule:      Suppose two vectors P and Q are drawn from the same origin. A straight line is drawn parallel to P and another straight line parallel to P and another straight line parallel to Q so as to form  parallelogram. The Resultant R = P+Q is given by diagonal of the parallelogram.

2) Triangular Method:-        Suppose two vector P and Q are drawn sucn inar tail of vector Q touches the head of vector P. the resultant R = P + Q is obtained at third side of the triangle.

3. Polygon Method: ( Head to tail Rule) Suppose there are four vectors A,     B,    C,    D in this method of vector addition, the tai of the 2^nd vector B is placed on the head of the Q touches the head of the 1^st vector A. similarly tail of 3^rd vector C is placed on head of 2^nd vector B and so on. The resultant vector R is obtained by joining tail of 1^st vector A with head of last vector. This is also called head to tail rule.

Since 5this complete a polygon therefore this is also called Polygon methods.

SUBTRACTION OF VECTORS

Suppose there are two vectors A and B

If we want to subtract B form A, we must take – ve of the vector B first and then add – B into A.

Subtraction of a vector is equivalent to the addition of the same vector with its direction reversed. Thus to subtract vector B for Vector A, reverse the direction of B and add it to A as shown.

Subtraction = A  +  (-B)

= A – B

RESOLUTION OF A VECTOR:-

The splitting of a vector into its rectangular components is called Resolution of a vector. it is reverse of vector addition.

Component of a Vector:-  A component of a vector is its effective value in a given direction.

Rectangular Components:-          Two components of a vector which are at right angle to each other are called Rectangular components.

One of the component is along horizontal axis called X-component.

One of the components is along vertical axis called y-component.

Suppose A has tow components.

(i) Ax = A Cos θ

                        (ii) Ay = A Sin  θ

A = A_x + A_y

TO FIND A VECTOR FORM RECTANGULAR COMPONENTS:-

Pathagour’s theorem is applied.

(Hyp)² = (Base)² + (Prep)²

(A)²  = (Ax)² + (Ay)²

Ax  = Ax² + Ay²

A  =  Ax² + Ay²

DIRECTION OF VECTOR:-

                                                Tan θ  =          Ay

Ax

θ          =          Tan^-1   (Ay)

Ax

MULTIPLICATION OF VECTORS:-

There are two types of multiplication 9or product) of vectors.

• Scalar Product
• Vector Product

SCALAR PRODUCT:-

When the product of two vector results into a scalar quantity, the product is called scalar product.

Scalar product is denoted by putting a do (.) between the two vectors to be multiplied therefore it is also called Dot product.

Value of scalar Product:

If there are two vectors A and B making an angle θ with each other then their scalar product is given by:-

A  .  B   = AB Cos θ

2^nd Definition:-

Product of magnitude of 1^st vector and magnitude of component of 2^nd vector in the direction of 1^st vector is called scalar product.

OR

                        Product of magnitude of 1^st vector and magnitude of x-component of 2^nd vector is called Scalar product.

(e.g)                                        A  .  B  =  AB Cos θ

A  .  B  =  A (Bx)

Examples:-

(i)        (Force)           (Displacement)                     = Work

(ii)       (Force)           (Velocity)                              = Power

(iii)     (Electric Intensity) (Area)                           = Electric Flux

TORQUE

• Physical quantity which produces angular acceleration in a body is called

OR

• Torque is cause of rotational motion.

OR

• Turning effect of a given force about the axis of rotation is called

Mathematical Definition:-

• Cross product of force and position vector is called e.

(i.e)                 T =

This shows that Torque is a vector quantity.

Direction of Torque:-         As torque is vector quantity so it must have direction its direction is determined by right hand rule which shows that torque is perpendicular to plane containing r and F

Factors upon which Torque Depends:

• Position vector r of the point about which the torque is measured.

Moment Arm:-        Perpendiculer distance of line of action of force from axis of rotation (origion) is called

Torque = Fr Sinθ

= Force  moment arm

= Moment of Force

Torque:-        Moment of force is called Torque.

OR

Product of force and moment arm is called

Special Cases:-

Case 1:-         If θ = 0, then              T = rF Sin0 = 0

Case 2:-         If θ = 180, then         T = rF Sin180 = 0

Case 3:-         If θ = 90, then           T = rF Sin90 = rF

Case 4:-         If the body is at rest or rotating with uniform angular velocity the angular acceleration will be zero. In this case the torque acting on the body will be zero.

TYPES OF TORQUE

Positive Torque:-                Anticlockwise torque is taken as positive.

Negative Torque:-              clockwise torque is taken as negative.

S.I Unit of Torque:-           N-m

Dimensions:-                        Torque = N-m

Torque = Ma  m

Torque = Kg m