# Unit and measurement

A. Short questions

1.

Dimensional analysis has following limitations:

1. Through dimensional analysis we cannot find the values of dimensionless constants. Consider the example of derivation of kinetic energy through dimensional analysis,

${\rm{K}}.{\rm{E}} \propto {{\rm{m}}^{\rm{x}}}{{\rm{v}}^{\rm{y}}}$

${\rm{K}}.{\rm{E}} = {\rm{k\: }}{{\rm{m}}^{\rm{x}}}{{\rm{v}}^{\rm{y}}}$

Converting in dimensional form

${\rm{J}} = {\rm{k\: k}}{{\rm{g}}^{\rm{x}}}{\left( {{\rm{m}}{{\rm{s}}^{ - 1}}} \right)^{\rm{y}}}$

${\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}} = {\rm{k\: }}{{\rm{M}}^{\rm{x}}}{{\rm{L}}^{\rm{y}}}{{\rm{T}}^{ - {\rm{y}}}}$

Comparing powers of dimensions we get

 Comparing M 1=x Equation – 1 Comparing L 2=y Equation – 2 Comparing T -2=-y Equation – 3

Using values obtained through the above equations, the kinetic energy can be written as,

${\rm{K}}.{\rm{E}} = {\rm{k\: }}{{\rm{m}}^1}{{\rm{v}}^2}$

We have derived the correct relationship between the physical quantities but still the value of dimensionless ‘k’ is unknown.

2.

The principle of homogeneity of dimensions is used to verify the physical laws involving physical quantities. For a relation to be valid and correct the dimensional form of the physical law must be balanced, i.e. the dimensions at both side of the equation must be same. Every physical law that is valid obeys the principle of homogeneity of dimensions.

For example consider following equations:

Equations of motion:

${{\rm{v}}_{\rm{f}}} = {{\rm{v}}_{\rm{i}}} + {\rm{a\: t}}$

${\rm{L}}{{\rm{T}}^{ - 1}} = {\rm{L}}{{\rm{T}}^{ - 1}} + {\rm{L}}{{\rm{T}}^{ - 2}}{\rm{\: T}}$

${\rm{L}}{{\rm{T}}^{ - 1}} = {\rm{L}}{{\rm{T}}^{ - 1}} + {\rm{L}}{{\rm{T}}^{ - 1}}$

${\rm{s}} = {{\rm{v}}_{\rm{i}}}{\rm{\: t}} + \frac{1}{2}{\rm{a\: }}{{\rm{t}}^2}$

${\rm{L}} = {\rm{L}}{{\rm{T}}^{ - 1}}{\rm{\: T}} + \frac{1}{2}{\rm{L}}{{\rm{T}}^{ - 2}}{{\rm{T}}^2}$

${\rm{L}} = {\rm{L}} + \frac{1}{2}{\rm{L}}$

$2{\rm{\: a\: s}} = {\rm{v}}_{\rm{f}}^2 - {\rm{v}}_{\rm{i}}^2$

$2{\rm{\: L}}{{\rm{T}}^{ - 2}}{\rm{\: L}} = {\left( {{\rm{L}}{{\rm{T}}^{ - 1}}} \right)^2} - {\left( {{\rm{L}}{{\rm{T}}^{ - 1}}} \right)^2}$

$2{{\rm{L}}^2}{{\rm{T}}^{ - 2}} = {{\rm{L}}^2}{{\rm{T}}^{ - 2}} - {{\rm{L}}^2}{{\rm{T}}^{ - 2}}$

Mass – Energy equivalence:

${\rm{E}} = {\rm{m}}.{{\rm{c}}^2}$

${\rm{J}} = {\rm{kg}}.{\left( {{\rm{m}}{{\rm{s}}^{ - 1}}} \right)^2}$

${\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}} = {\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}$

All the equations that are given above are dimensionally correct i.e. the dimensions are same at both sides of the equation. It is to be noted that all the constants are ignored in dimensional form and also dimensions are never to be canceled by subtraction, just like we have two similar terms with opposite signs in last equation of motion.

3.

No, dimensional analysis does not tell that a physical relation is completely right because numerical factors in the relation cannot be determined.

4.

Dimensionless quantities are those physical quantities which do not have any physical dimension associated with them. These usually result as a product or ratio of physical quantities such that their dimensions get cancelled out leaving a dimensionless entity.

Examples are:

Strain – ratio of change in length and original length

${\rm{strain}} = \frac{{\Delta {\rm{l}}}}{{\rm{l}}} = \frac{{\rm{L}}}{{\rm{L}}}$

Refractive index – ratio of speed of light in vacuum and speed of light in a medium

${\rm{refractive\: index}} = \frac{{\rm{c}}}{{\rm{v}}} = \frac{{{\rm{L}}{{\rm{T}}^{ - 1}}}}{{{\rm{L}}{{\rm{T}}^{ - 1}}}}$

Although the dimension less quantities doesn’t have dimensions but they can have unit associated with them. These units are termed as dimension less units.

5.

Yes we can tell the most unit of physical quantity from its dimensions.

6.

If a body moves with uniform velocity v for time t, the distance covered x is given by x = vt. Now, if x is measured in meter, vt should also be in meter. This is known as unit consistency.

7.

π is unit less quantity.

8.

The principle of homogeneity of dimensions is used to verify the physical laws involving physical quantities. For a relation to be valid and correct the dimensional form of the physical law must be balanced, i.e. the dimensions at both side of the equation must be same. Every physical law that is valid obeys the principle of homogeneity of dimensions.

9.

The principle of homogeneity of dimensions is used to verify the physical laws involving physical quantities. For a relation to be valid and correct the dimensional form of the physical law must be balanced, i.e. the dimensions at both side of the equation must be same. Every physical law that is valid obeys the principle of homogeneity of dimensions.

For example consider following equations:

Equations of motion:

${{\rm{v}}_{\rm{f}}} = {{\rm{v}}_{\rm{i}}} + {\rm{a\: t}}$

${\rm{L}}{{\rm{T}}^{ - 1}} = {\rm{L}}{{\rm{T}}^{ - 1}} + {\rm{L}}{{\rm{T}}^{ - 2}}{\rm{\: T}}$

${\rm{L}}{{\rm{T}}^{ - 1}} = {\rm{L}}{{\rm{T}}^{ - 1}} + {\rm{L}}{{\rm{T}}^{ - 1}}$

${\rm{s}} = {{\rm{v}}_{\rm{i}}}{\rm{\: t}} + \frac{1}{2}{\rm{a\: }}{{\rm{t}}^2}$

${\rm{L}} = {\rm{L}}{{\rm{T}}^{ - 1}}{\rm{\: T}} + \frac{1}{2}{\rm{L}}{{\rm{T}}^{ - 2}}{{\rm{T}}^2}$

${\rm{L}} = {\rm{L}} + \frac{1}{2}{\rm{L}}$

$2{\rm{\: a\: s}} = {\rm{v}}_{\rm{f}}^2 - {\rm{v}}_{\rm{i}}^2$

$2{\rm{\: L}}{{\rm{T}}^{ - 2}}{\rm{\: L}} = {\left( {{\rm{L}}{{\rm{T}}^{ - 1}}} \right)^2} - {\left( {{\rm{L}}{{\rm{T}}^{ - 1}}} \right)^2}$

$2{{\rm{L}}^2}{{\rm{T}}^{ - 2}} = {{\rm{L}}^2}{{\rm{T}}^{ - 2}} - {{\rm{L}}^2}{{\rm{T}}^{ - 2}}$

Mass – Energy equivalence:

${\rm{E}} = {\rm{m}}.{{\rm{c}}^2}$

${\rm{J}} = {\rm{kg}}.{\left( {{\rm{m}}{{\rm{s}}^{ - 1}}} \right)^2}$

${\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}} = {\rm{M}}{{\rm{L}}^2}{{\rm{T}}^{ - 2}}$

All the equations that are given above are dimensionally correct i.e. the dimensions are same at both sides of the equation. It is to be noted that all the constants are ignored in dimensional form and also dimensions are never to be canceled by subtraction, just like we have two similar terms with opposite signs in last equation of motion.

10.

In the equation C = A + B the dimensions of C are equal to the dimensions of A + B.

11.

1. It is a rational system of units

2. It is a coherent system of units

3. It is a decimal system of units

12.

Precision is determined by the least count of the measuring instrument. The smaller the least count, the greater is the precision.

13.

The number of meaningful digits in number is called its number of significant figures.

14.

Reshaping the measure of a physical quantity with the least deviation from its original value after dropping the last digits which are not required is called rounding off. It is necessary for Reshaping the measure of a physical quantity with the least deviation from its original value after dropping the last digits which are not required.

15.

The accuracy of measurement means the extent to which a measure value agrees with the standard or true for the measurement.

Precision of measurement means the extent to which a given set of measurement of the same quantity agree with their mean value.

16.

a. Power- [M L2 T-3]

b. Force- [M L T-2]

c. Gravitational constant-[M-1 L3 T-2]

d. Acceleration due to gravity-[M0 L T-2]

e. Torque-[M L2 T-2]

17.

a. Dimensional variables: Physical quantities which have dimensions but do not have fixed value are called dimensional variables, eg; work, power, velocity.

b. Dimensionless variables: These are physical quantities which have neither dimensions nor fixed value.

c. Dimensional constant: These are physical quantities which have dimensions and fixed value.

d. Dimensionless constant: Physical quantities which do not posses dimensions but have fixed value are called dimensional constants, eg; π, counting number etc.

1.

The mathematical relation contains many parameters concerning the phenomenon. The various parameters appearing in the relation are called physical quantities.

 Fundamental Unit Derived unit The unit of the fundamental quantities are called fundamental unit. The units of the derived quantities are called derived unit. The fundamental unit depend upon the system of unit used. The derived unit depend upon the system of units taken. Eg: Kilogram,metre etc Eg: Unit of force, Newton etc

2.

The dimension of a physical quantity is definaed as the power to be raised to the fundamental quantities in order to represent the physical quantity.

The dimensional formula is defined as the expression of the physical quantity in terms of its basic unit with proper dimensions.

Eg:

Force=$\frac{{{\rm{Work\: done}}}}{{{\rm{Time}}}}$

$= \frac{{{\rm{Force*distance}}}}{{{\rm{Time}}}}$

=$\frac{{{\rm{Mass*Accleration*Distance}}}}{{{\rm{Time*TIme*Time}}}}$

=$\frac{{\left[ {\rm{M}} \right]{\rm{*}}\left[ {\rm{L}} \right]{\rm{*}}\left[ {\rm{L}} \right]}}{{\left[ {\rm{T}} \right]{\rm{*}}\left[ {\rm{T}} \right]{\rm{*}}\left[ {\rm{T}} \right]}}$

=$\frac{{\left[ {\rm{M}} \right]{\rm{*}}{{\left[ {\rm{L}} \right]}^3}}}{{{{\left[ {\rm{T}} \right]}^3}}}$

=[ML2T-3]

Thus, Power (P) =[ML2T-3]

Hence, the dimension of power is 1 in mass, 2 in length and -3 in time.

Some dimensional formula of some physical quantities

 Distance Length [L] Area Length * Width [L2] Velocity $\frac{{{\rm{Distance}}}}{{{\rm{Time}}}}$     [MLT-1] Gravitational Constant (G) $\frac{{{\rm{Force*}}{{\left( {{\rm{Distance}}} \right)}^2}}}{{{{\left( {{\rm{Mass}}} \right)}^2}}}$  [M-1L3T-2] Impulse Force * Time [MLT-1]

An equation explaining physical quantities with dimensional formula is known as dimensional equation.

For example, the equation ${\rm{S}} = {\rm{ut}} + \frac{1}{2}{\rm{a}}{{\rm{t}}^2}$ represents the physical equation for the distance travelled by a body accelerating of acceleration (a) for time (t) with initial velocity (u). Then, replacing u,a,s and t by their dimensional formula, we get,

S = [M0-LT1]*[M0L0T]+[M0LT-2]*[M0L0T-2]

This equation is called dimensional equation.

3.

The main uses of dimensional equations are:

a. To check the correctness of a physical equation.

b. To convert a system of unit to another system of unit.

c. To determine the dimension of a constant in a relation

d. To derive the relation between various physical quantities.

e. To correct a formula

4.

A unit is defined as a convention to define an amount of physical property in a specific system of units. There are several systems of units with different conventions to express these units. There are units defined for every physical quantity and that physical quantity is expressed in terms of that specific unit. For example, in International System of units we have kilogram for mass, meter for length, seconds/minutes/hours for time, etc. Without a unit a physical property cannot be distinguished or described.

For writing, units are expressed in terms of alphabets, referred to as unit symbols. Like meter is written ‘m’, second as‘s’, grams as ’g’, etc.

To expresses larger quantities of units in terms of power of tens, certain prefixes are used.

Examples: kilogram for 1000 grams, centimeters for 1/100 of a meter meters, milliseconds for 1/1000 of a second, etc.

The Si system has seven fundamental physical quantities given below:

 Physical Quantity Unit Symbol mass kilogram kg length Meter m time second s temperature Kelvin K electric current ampere A amount of matter Mole mol luminous intensity candela Cd

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