#### Study Material and Notes of Ch 2 Units and Measurements Class 11th Physics

**Topics in the Chapter**Â

â€¢ Physical Quantity

â€¢ Fundamental and Derived Units

â†’ S.I. System

â†’ Dimensions of a Physical Quantity

â€¢ Important Dimensions of Complete Physics

â€¢ Application of Dimensional Analysis

â†’ Limitations of Dimensional Analysis

â€¢ Significant Figures

â€¢ Rounding off

â€¢ Significant Figures in Calculation

â€¢ Order of Magnitude

â€¢ Errors of Measurement

â€¢ Propagation of Errors

**Physical Quantity**

â†’ A quantity which can be measured and expressed in form of laws is called a physical quantity.

Physical quantity (Q) = Magnitude Ã— Unit = n Ã— u

where,

n = numerical value

u = unit.

As the unit(u) changes, the magnitude (n) will also change but product â€˜nuâ€™ will remain same.

i.e. nu = constant, or n

_{1}u_{1}Â =Â n_{2}u_{2}Â = constant**Fundamental and Derived Units**

Any unit of mass, length and time in mechanics is called a fundamental, absolute or base unit. Other units which can be expressed in terms of fundamental units, are called derived units.

**System of units**: A complete set of units, both fundamental and derived for all kinds of physical quantities is called system of units.

There are four systems of units. We only have to study the S.I. system.

(1) CGS system

(2) MKS system

(3) FPS system.

(4) S.I. system

**S.I. System**

It is known as International system of units. There are seven fundamental quantities in this system.Â

**Definitions of fundamental quantities**

**(i) Metre:**The metre is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second.

**(ii) Kilogram:**The kilogram is equal to the mass of the international prototype of the kilogram (a platinum-iridium alloy cylinder) kept at international Bureau of Weights and Measures, at Sevres, near Paris, France.

**(iii) Second:**The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.

**(iv) Ampere:**Â The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum,

would produce between these conductors a force equal to 210

^{â€“7}newton per metre of length.**(v) Kelvin:**Â The kelvin, is the fraction 1/273.16 of the thermodynamicÂ temperature of the triple point of water.

**(vi) Mole:**The mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12.

**(vii) Candela:**Â The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 54010

^{12}hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

Besides the above seven fundamental units two supplementary units are also defined - Radian (rad) for plane angle and Steradian (sr) for solid angle.

**Dimensions of a Physical Quantity**

When a derived quantity is expressed in terms of fundamental quantities, it is written as a product of different powers of the fundamental quantities.

The powers to which fundamental quantities must be raised in order to express the given physical quantity are called its dimensions.

**Important Dimensions of Complete Physics**

**Mechanics**

**Heat**

**Quantities Having Same Dimensions**

**Application of Dimensional Analysis.**

(i) To find the unit of a physical quantity in a given system of units.

(ii) To find dimensions of physical constant or coefficients.

(iii) To convert a physical quantity from one system to the other.

(iv) To check the dimensional correctness of a given physical relation: This is based on the â€˜principle of homogeneityâ€™. According to this principle the dimensions of each term on both sides of an equation must be the same.

(v) To derive new relations.

**Limitations of Dimensional Analysis.**

(i) If dimensions are given, physical quantity may not be unique.

(ii) Numerical constant having no dimensions cannot be deduced by the methods of dimensions.

(iii) The method of dimensions can not be used to derive relations other than product of power functions.

For example,

s = ut + (1/2) at

^{2}Â or y = a sin Ï‰t
(iv) The method of dimensions cannot be applied to derive formula consist of more than 3 physical quantities.

**Significant Figures**

Significant figures in the measured value of a physical quantity tell the number of digits in which we have confidence.

Larger the number of significant figures obtained in a measurement, greater is the accuracy of

the measurement. The reverse is also true.

The following rules are observed in counting the number of significant figures in a given measured quantity.

(i) All non-zero digits are significant.

(ii) A zero becomes significant figure if it appears between two non-zero digits.

(iii) Leading zeros or the zeros placed to the left of the number are never significant.

__Example:__

0.543 has three significant figures.

0.006 has one signifi cant fi gures.

(iv) Trailing zeros or the zeros placed to the right of the number are significant.

__Example:__

4.330 has four significant figures.

343.000 has six significant figures.

(v) In exponential notation, the numerical portion gives the number of significant figures.

__Example:__

1.32 Ã— 10

^{-2}Â has three significant figures.**Rounding Off**

(1) If the digit to be dropped is less than 5, then the preceding digit is left unchanged.

__Example:__

x = 7.82 is rounded off to 7.8,

again x = 3.94 rounded off to 3.9.

(2) If the digit to be dropped is more than 5, then the preceding digit is raised by one.

__Example:__

x = 6.87 is rounded off to 6.9, again x = 12.78 is rounded off to 12.8.

(3) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.

__Example:__

x = 16.351 is rounded off to 16.4, again x = 6.758 is rounded off to 6.8.

(4) If digit to be dropped is 5 or 5 followed by zeros, then preceding digit is left unchanged, if it is even.

__Example:__

x = 3.250 becomes 3.2 on rounding off, again x = 12.650 becomes 12.6 on rounding off.

(5) If digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd.

__Example:__

x = 3.750 is rounded off to 3.8, again x = 16.150 is rounded off to 16.2.

**Significant Figures in Calculation**

The following two rules should be followed to obtain the proper number of significant figures in any calculation.

(i) The result of an addition or subtraction in the number having different precisions should be reported to the same number of decimal places as are present in the number having the least number of decimal places.

(ii) The answer to a multiplication or division is rounded off to the same number of significant figures as is possessed by the least precise term used in the calculation.

**Order of Magnitude**

Order of magnitude of quantity is the power of 10 required to represent the quantity. For determining this power, the value of the quantity has to be rounded off.

While rounding off, we ignore the last digit which is less than 5. If the last digit is 5 or more than five, the preceding digit is increased by one.

**Example:**

(1) Speed of light in vacuum = 3Ã—10

^{8}ms^{-1}Â â‰ˆ10^{8}Â m/s (as 3 < 5)
(2) Mass of electron = 9.1 Ã— 10

^{-31}Â kg 10^{-30}Â â‰ˆ kg (as 9.1 > 5).**Errors of Measurement.**

The measured value of a quantity is always somewhat different from its actual value, or true value.

This difference in the true value of a quantity is called error of measurement.

**(i) Absolute error:**Absolute error in the measurement of a physical quantity is the magnitude of the difference between the true value and the measured value of the quantity.

Let a physical quantity be measured n times. Let the measured value be a

_{1}, a_{2}, a_{3}, . . . a_{n}. The arithmetic mean of these value is am = (a_{1}Â + a_{2}Â + a_{3})/n
Usually, a m is taken as the true value of the quantity, if the same is unknown otherwise.

By defi nition, absolute errors in the measured values of the quantity are

â–³a

_{1}Â = a_{m}Â â€“ a_{1}
â–³a

_{2}Â = a_{m}Â â€“ a_{1}
. . . . . . . . . . .

â–³a

_{n}Â = a_{m}Â â€“ a_{2}
The absolute errors may be positive in certain cases and negative in certain other cases.

**(ii) Mean absolute error:**It is the arithmetic mean of the magnitudes of absolute errors in all the measurements of the quantity. It is represented by â–³a. Thus

Hence, the final result of measurement may be written as a = am Â±â–³a. This implies that any measurement of the quantity is likely to lie between am -â–³a and am +â–³a.

**(iii) Relative error or Fractional error:**

Relative error or Fractional errorÂ

**(iv) Percentage error**

Percentage errorÂ

**Propagation of Errors**

**(i) Error in sum of tie quantities:**Suppose x = a + b

Let â–³a = absolute error in measurement of a

â–³b = absolute error in measurement of b

â–³x = absolute error in calculation of x i.e. sum of a and b.

The maximum absolute error in x is â–³x = Â± (â–³a + â–³b)

**(ii) Error in difference of the quantities:**Suppose x = a â€“ b

The maximum absolute error in x is â–³x = Â± (â–³a + â–³b)

**(iii) Error in product of quantities:Â**Suppose x = a Ã— b

The maximum fractional error in x is â–³x/x = Â±(â–³a/aÂ +Â â–³b/a)

**(iv) Error in division of quantities:**Suppose x = a/b

The maximum fractional error in x is â–³x/x = Â±(â–³a/aÂ +Â â–³b/a)

**(v) Error in quantity raised to some power:**Suppose x = a

^{n}/b

^{m}

The maximum fractional error in x is â–³x/x = Â±(nâ–³a/aÂ + mâ–³b/a)

The quantity which have maximum power must be measured carefully because itâ€™s contribution to error is maximum.