Chapter - 2 Units and measurements

The comparison of any physical quantity with its standard unit is called measurement.

Physical Quantities

All the quantities in terms of which laws of physics are described, and whose measurement is necessary are called physical quantities.

Units

A definite amount of a physical quantity is taken as its standard unit.
The standard unit should be easily reproducible, internationally accepted.

Fundamental Units

Those physical quantities which are independent to each other are called fundamental quantities and their units are called fundamental units.

S.No. Fundamental Quantities Fundamental Units Symbol

1. Length metre - m
2. Mass kilogram - kg
3. Time second - S
4. Temperature kelvin - K
5 Electric current ampere - A
6 Luminous intensity candela - cd
7 Amount of substance mole - mol

Supplementary Fundamental Units

Radian and steradian are two supplementary fundamental units. It measures plane angle and solid angle respectively.

S.No. Supplementary Fundamental Quantities Supplementary Unit Symbol

1 Plane angle radian - rad

2 Solid angle steradian - Sr

Derived Units

Those physical quantities which are derived from fundamental quantities are called derived quantities and their units are called derived units.

e.g., velocity, acceleration, force, work etc.

Definitions of Fundamental Units

The seven fundamental units of SI have been defined as under.

1. 1 kilogram A cylindrical prototype mass made of platinum and iridium alloys of height 39 mm and diameter 39 mm. It is mass of 5.0188 x 1025 atoms of carbon-12.

2. 1 metre 1 metre is the distance that contains 1650763.73 wavelength of orange-red light of Kr-86.

3. 1 second 1 second is the time in which cesium atom vibrates 9192631770 times in an atomic clock.

4. 1 kelvin 1 kelvin is the (1/273.16) part of the thermodynamics temperature of the triple point of water.

5. 1 candela 1 candela is (1/60) luminous intensity of an ideal source by an area of cm’ when source is at melting point of platinum (1760°C).

6. 1 ampere 1 ampere is the electric current which it maintained in two straight parallel conductor of infinite length and of negligible cross-section area placed one metre apart in vacuum will produce between them a force 2 x 10-7 N per metre length.

7. 1 mole 1 mole is the amount of substance of a system which contains a many elementary entities (may be atoms, molecules, ions, electrons or group of particles, as this and atoms in 0.012 kg of carbon isotope 6C.

Systems of Units

A system of units is the complete set of units, both fundamental and derived, for all kinds of physical quantities. The common system of units which is used in mechanics are given below:

1. CGS System In this system, the unit of length is centimetre, the unit of mass is gram and the unit of time is second.

2. FPS System In this system, the unit of length is foot, the unit of mass is pound and the unit of time is second.

3. MKS System In this system, the unit of length is metre, the unit of mass is kilogram and the unit of time is second.

4. SI System This system contain seven fundamental units and two supplementary fundamental units.

Relationship between Some Mechanical SI Unit and Commonly Used Units

S.No. Physical Quantity Unit

1. Length (a) 1 micrometre = 10-6 m

(b) 1 angstrom =10-10 m

2. Mass (a) 1 metric ton = 103kg

(b) 1 pound = 0.4537 kg

3 Volume. 1 litre = 10-32 m³

4. Force. (a) 1 dyne = 10-5 N

(b) 1 kgf = 9.81 N

5. Pressure. (a) 1 kgfm2= 9.81Nm-2

(b) 1 mm of Hg = 133 Nm-2

(d) 1 atmosphere pressure = 76 cm of Hg = 1.01 x 105pascal

6. Work and energy (a) 1 erg =10-7J

(b) 1 kgf-m = 9.81 J

(d) 1 eV = 1.6 x 10-19 J

7. Power. (d) 1 kgf- ms-1= 9.81W1 horse power = 746 W

Some Practical Units

1. 1 fermi =10-15 m

2. 1 X-ray unit = 10-13 m

3. 1 astronomical unit = 1.49 x 1011 m (average distance between sun and earth)

4. 1 light year = 9.46 x 1015 m

5. 1 parsec = 3.08 x 1016 m = 3.26 light year

Some Approximate Masses

Object Kilogram

Our galaxy 2 x 1041

Sun 2 x 1030

Moon 7 x 1022

Asteroid Eros 5 x 1015

Dimensions

Dimensions of any physical quantity are those powers which are raised on fundamental units to express its unit. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity, is called the dimensional formula.

Homogeneity Principle

If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle.

Mathematically [LHS] = [RHS]

Applications of Dimensions

1. To check the accuracy of physical equations.

2. To change a physical quantity from one system of units to another system of units.

3. To obtain a relation between different physical quantities.

Significant Figures

In the measured value of a physical quantity, the number of digits about the correctness of which we are sure plus the next doubtful digit, are called the significant figures.

Rules for Finding Significant Figures

1. All non-zeros digits are significant figures, e.g., 4362 m has 4 significant figures.

2. All zeros occuring between non-zero digits are significant figures, e.g., 1005 has 4

significant figures.

3. All zeros to the right of the last non-zero digit are not significant, e.g., 6250 has only 3

significant figures.

4. In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00325 has only 3 significant figures.

5. All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.4750 has 5 significant figures.

Significant Figures in Algebric Operations

(i) In Addition or Subtraction In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. e.g.,

If l1= 4.326 m and l2 = 1.50 m

Then, l1 + l2 = (4.326 + 1.50) m = 5.826 m

As l2 has measured upto two decimal places, therefore

l1 + l2 = 5.83 m

(ii) In Multiplication or Division In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values. e.g., If length

1= 12.5 m and breadth b = 4.125 m.

Then, area A = l x b = 12.5 x 4.125 = 51.5625 m2

As l has only 3 significant figures, therefore

A= 51.6 m2

Rules of Rounding Off Significant Figures

1. If the digit to be dropped is less than 5, then the preceding digit is left unchanged. e.g., 1.54 is rounded off to 1.5.

2. If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g., 2.49 is rounded off to 2.5.

3. If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g., 3.55 is rounded off to 3.6.

4. If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g., 3.750 is rounded off to 3.8 and .650 is rounded off to 4.6.

Error

The uncertainty in the measurement of a physical quantity is called an error.The errors in measurement can be classified as

(i) Systematic errors and

(ii) Random errors

SYSTEMATIC ERRORS:

These are the errors that tend to be either positive or negative.Sources of systematic errors are

(i) Instrumental errors

(ii) Imperfection in experimental technique or procedure

(iii) Personal errors

RANDOM ERRORS :

Those errors which occur irregularly .These errors arise due to unpredictable fluctuations in experimental conditions

Least count error

Least count error is the error associated with the resolution of the instrument.

Absolute error

The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement.

Mean Absolute Error

The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by Δa_mean

_{Relative error - it is the ratio of the mean absolute error to the true value.}

_{Relative error = Δa mean/ a mean}

_{Percentage Error : When the relative error is expressed in per cent, it is called the percentage error (δa).}

_{Percentage error =( Δa mean/ a mean) ×100}

_{COMBINATION OF ERRORS}

_{ERROR OF A SUM OR A DIFFERENCE}

_{When two quantities are added or subtracted, the absolute error in the final result is thesums of the absolute errors in the individual quantities.}

_{IF Z=A+ B then the max possible error in Z, ∆Z =∆A + ∆B}

_{IF Z=A- B then the max possible error in Z, ∆Z =∆A + ∆B}

_{ERROR OF A PRODUCT OR A QUOTIENT}

_{When two quantities are multiplied or divided the relative error is the sum of the relative errors in the multipliers}

_{Suppose Z= A*B or Z=A/B then the max relative error in ‘Z’ = ∆Z/Z= (∆A/A) + (∆B/B)}

_{ERROR IN CASE OF A QUANTITY RAISED TO A POWER}

_{The relative error in a physical quantity raised to the power k is the k times the relative}

_{error in the individual quantity.}

_{Suppose Z = Ak}

_{then ∆Z/Z = K (∆A/A)}

Chapter 2 units and measurements