Comprehensive WBBSE Class 9 Physical Science Notes Chapter 1 Measurement can help students make connections between concepts.

## Measurement Class 9 WBBSE Notes

Science : This word originated from Latin word ‘Scientia’ meaning ‘to know’. Thus knowledge acquired by man through systemetic observations and experiments is called Science.

Physical Quantity : A physical quantity is any measurable quantity of an object or an event. Example : mass, length, time, weight, density etc.

Types of physical quantity :

- Scalar quantity
- Vector quantity

Scalar quantity : The quantity which has only magnitude but no direction is called scalar quantity.

Example : mass, length, time etc.

Vector quantity : The quantity which has both magnitude and direction is called vector quantity.

Example : weight of matter, velocity, acceleration, force etc.

Unit : In measuring any physical quantity, some convenient and definite quantity of it is taken as the standard and in terms of this standard the physical quantity is measured. This standard is called a unit.

Importance of units :

- Measurement of any physical quantity without unit is meaningless. Because we cannot have any idea about a physical quantity with its magnitude only.
- Unit establishes relation between different measures of same quantity.

Characteristics of units :

- The unit should be well defined.
- The unit should be of suitable size.
- The unit should be easily reproducible.
- The unit should be imperishable.
- The unit should not change with time or with physical conditions like pressure, temperature etc.

Fundamental unit : This units of physical quantities which are independent of each other and from which other units can be derived are called the fundamental units.

Example : units of length, mass, time etc.

Derived unit : The units of physical quantity which are derived with the help of one or more than one fundamental units are called derived units.

Example : The unit of area is obtained by using the unit of length twice. Similarly units of speed, force, work etc. are all derived units.

Different systems of fundamental units :

- CGS system : The word ‘C’ is the unit of length-Centimeter, ‘G’ is the unit of mass gram, ‘S’ is the unit of time-Second.
- EPS system : This is the British system of units in which units of length, mass and time are foot, pound and second respectively.
- MKS system : This is basically practical systems of units in which units of length, mass and time are metre, kilogram and second respectively.

SI units : In 1960, an international system of units was adopted to have a consistent system of units. This system of units is known as SI units.

Types of SI units :

- Fundamental SI units
- Derived SI units

Fundamental SI Units :

Derived SI units :

Some important rules for writing SI units :

- The symbol of the units are always written in Roman small letter. Example : {kg}, {m}, {s} etc.
- If a unit is named after a person, the symbol is written in capital letter. Example : Newton-N, Ampere-A, Volt-V etc.
- The symbols of the units are always used in singular form. Example : Mass- 5 kg not 5 kgs.
- When temperature is expressed in Kelvin scale, (0°) degree sign is not used i.e. we write 273 K, not 273°k
- Full stop, comma, etc. are not written after the symbols of the units. (e.g. we should write cm and not cm. or cm,)
- The multiplication of two units are written as symbols of unit sequencially, e.g. kgm.
- units like metre per second is written as m/s or ms
^{-1}or in case of Joule per kelvin per mole is written as JK^{-1}mol^{-1}or J/K mol but not J/K/mol.

Definitions of units of SI :

Metre : The distance between the two marks made on a platinumiridium bar maintained at 0°C temperature preserved at International Bureau of Weights and Measures (Bureau international des poids et measures) at Sevres near Paris considered as one metre (symbol : m).

Modern definition of metre : The standard metre is exactly equal to 1650763.73 wavelengths in vacuum, of the radiation from Krypton isotope of mass number 86.

Kilogram : Mass of a solid cylinder made of platinum-iridium and preserved at the standard office at Sevres near Paris, is called one kilogram (kg).

Modern definition of second : One second is the duration of 9192631770 periods of radiation corresponding to unperturbed transition between the two hyperfine levels of the ground state of \({ }^{193} {C}_{{s}}\) atom.

Units of volume :

Volume : It is defined to be the space occupied by a substance.

Solids have three dimensions, i.e. length, breadth and height and so the unit of volume is CGS system in cm × cm × cm = cm^{3}.

Unit of volume in SI system is m^{3}.

Volume of some solid figures :

- Volume of a rectangle solid = length × breadth × height.
- Volume of a cube = (length)
^{3}. - Volume of a sphere = \(\frac{4}{3}\) πr
^{3} - Volume of a cone = \(\frac{1}{3}\) πr
^{2}h - Volume of a right circular cylinder = πr
^{2}h

[r = radius, π = \(\frac{22}{7}\), h = the perpendicular height]

Unit of volume of liquid :

Litre (L)= Volume of 1 kg of pure water at 4°C or 277 K is called a litre. This is not an SI unit. 1 L = 1000 ml = 1000 cm^{9} = 10 dm^{3}

Volume of 1 kg of water is 1000 ml.

Hence, volume of 1000 ml (cc) of pure water at 277 K is called a litre.

It is the unit of liquid in CGS system.

1 ml = 1 cc

Advantage of increased volume of water on solidification : In the winter season in the cold country water solidifies to ice and the volume of water increased on solidification. The density of ice being less than water it floats on the upper surface of water. Again by costing water reaches at the temperature of 4°C and it attains its maximum density and thereby it cannot come on the upper surface of water below the surface of ice and remain at 4°C. So aqua life is possible in winter.

Measurement of length : The measurement of lengths are of two types :

- direct method
- indirect method

Length measuring device in direct method :

- a metre scale : for 10-3 m-102 m length
- a Vernier – Callipers : for distances upto 10-4m
- a screw gauge or a spherometer : for distances upto 10-5 m.

Length measuring device in indirect method for large distance :

Parallax method : The change in the position of an object with respect to the background, when seen from two different positions is known as parallax. The distance between the two positions of observation is called the basis.

Size of an astronomical object : The size of an astronomical object, such as the moon can be measured with the help of an astronomical telescope.

Indirect methods for determination of very small length : The optical microscopes, working on visible light of wavelengths ranging from 4 × 10^{-7} m to 8 × 10^{-7} m, cannot be used to measure the sizes of molecules (10^{-8} m. to .10^{-16} m).

An electron microscope is usually used for this purpose.

Wavelength of radiations are expressed in angstrom (A°).

1 A°=10^{-8} cm = 10^{-10}m

Some commonly used prefixes in CGS and SI system :

Measurement of Mass :

The mass of a body is the quantity of matter contained in it. The range of mass varies from that of electron having mass of the order of 10-30 kg to observable universe with masses of the order of 1055 kg}.

Mass measuring device :

- The mass of ordinary objects-common balance
- The large masses of planets etc-gravitational methods are adopted.
- For measuring of very small masses of atomic or sub-atomic particles – mass spectrograph.

Few important points related to common balance :

If the mass of right-hand side pan and left-hand side pan in balance are unequal but same in length then the actual mass of substance

m = \(\frac{m_1+m_2}{2}\)

(where, m_{1} and m_{2} are mass of substance in two different pans)

If the length of balance is unequal but the pans are same in mass, then the actual mass of substance

m = \(\sqrt{m_1 m_2}\) (where m_{1} and m_{2} are mass in two pans)

Characteristics of Good balance :

- The pillar should be vertical, i.e. the balance beam horizontal with the help of levelling screws and plumb line.
- The balance should be correct.
- The balance should be rigid.
- The balance should be stable.

Different portion of comman balance :

- balance beam
- stirrups
- pointer
- lever
- scale pans
- plumb line
- glass case

Weight box : A wooden box for holding weights are called weight box is supplied each balance.

It may be noted that in a weight box weights one in the ratio 1 : 2 : 2 : 5.

A common balance can measure a minimum weight of 5 mg accurately. The upper range is about 200 g.

Spring balance: The weight of a body is the force with which is attracted by the earth towards its centre. The weight of a body is measured by spring balance.

Measurement of time : The range of time interval of events varies from as small as of the order 10-24 s} for life span of most unstable particle to as large as of the order of 1017 s} for age of universe.

For measurement of time interval, we need a clock based on any phenomenon that repeats itself regularly.

Commonly used clocks and watches are based on spring, pendulum etc.

Time measuring device :

Stop watch : It is used to measure the interval of incident starting and ending. This watch can measure a time interval of one-tenth of a second accurately. This watch is used in sports, scientific work etc. for measuring time.

Sand watch : This consists of two conical shaped glass vessels joined with each other with a narrow opening in the middle. The upper vessel contains a measured quantity of dry sand and it takes a definite interval of time for the sand to pass from the upper vessel to the lower one. The bottle is inverted for reuse when the upper vessel becomes empty.

Sundial : Sun rays are used in this time-measuring device. It consists of a horizontal circular disc with a thin triangular plate of metal fixed at its centre and pointing in the north-south direction. This triangular plate serves as an obstacle to the rays of the sun and casts shadow on the circular disc on the other side. The circular disc has graduations from 1 to 12 like that on a clock. Any particular time of the day, is indicated by the position of the shadow.

The sundial can be used only on a sunny day and not at night or on a cloudy day.

The pendulum clock : In this clock a simple pendulum is used. At a particular place, a pendulum of a given length takes fixed time to complete one oscillation i.e. starting from one extreme position and coming back to that position. This time is called the time period of the pendulum. A seconds pendulum is that whose time period is 2 seconds. In a pendulum clock such a seconds pendulum is used.

Atomic or caesium clock : These clocks are based on periodic vibrations produced in a caesium atom. These clocks are very accurate and in a year such a clock lose or gain not more than 3s.

Electric oscillators : The A.C. main electric supply has a frequency of 50 Hz. The synchronous rotations of an A.C. motor can be used for having a time scale.

Electronic oscillators : Vacuum tubes and transistors can be used for producing electro-magnetic waves of high frequencies and their small time periods of oscillations are used small time intervals accurately.

Quartz crystal clock : A quartz crystal shows piezo-electric effect. When fluctuating pressure is applied across one pair of faces of a crystal, an oscillating emf is developed across another pair perpendicular faces. These oscillations may be used for measuring time. It has an accuracy of 1 second in every 10^{-9}second.

Decay of elementary particles : Many unstable elementary particles decay in time interval as short as 10^{-10} second to 10^{-24} second. Thus studying their decay very small time intervals can be measured.

Radioactive dating : This technique is used for measuring long time interval of the order of 10^{-17} second.

Accuracy : The closeness of the measured value to the true value of the physical quantity is known as the accuracy of the measurement.

Precision : It means the extent or limit to which the measurement of a physical quantity is done.

Errors in measurement : The error in the measurement of a physical quantity is defined as the difference between the true value and the measured value of the physical quantity.

Types of errors :

- Systematic errors
- Random errors

Systematic errors : The error which occurs according to a definite pattern is known as systematic error.

Types of systematic errors :

- Instrumental errors : The errors caused due to defective instrument are called instrumental errors.
- Personal errors : The errors in the measurement of a physical quantity due to limitations or carelessness of the person experimenting are known as personal errors.
- Natural errors : The errors due to the change in the conditions of the environment (temperature, pressure etc.) are known as natural errors.

Random error : Random or chance errors are due to unknown causes. These errors cannot be controlled by the observer and are not constant in magnitude. They may be positive or negative.

The errors are expressed in different ways :

Absolute error: The absolute error of a given value of physical quantity is the difference between mean value of the physical quantity and the observed value under consideration.

Absolute error of the i-th observation

the mean value of the measured quantity,

x_{i} = the value of the measured quantity in the i th observation.]

Mean absolute error : The arithmetic mean or average of all absolute errors of the measurements is called the mean absolute error.

Mean absolute error

Relative error : The ratio of the absolute error to the physical quantity is called the relative error.

Relative error = Sk = \(\frac{\Delta \bar{x}}{x}\)

Percentage error = Relative error × 100%

= \(\frac{\Delta \bar{x}}{x}\) × 100%

Accuracy : The closeness of the measured value to the true value of the physical quantity is known as the accuracy of the measurement.

The degree of accuracy of any measurements depends upon :

the accuracy of the measuring device used.

Precision : It means the extent or limit to which the measurement of a physical quantity is done.

Significant figures : It is in the reported result of a measurement of a quantity is the number of digits that are known with certainty plus one that is uncertain, beginning with the first non-zero digit.

Rounding off : The observed results of various measurements may have different precisions. Thus, the results obtained at various stages of calculation are to be rounded off because the final result cannot be more precised than that of the least precised measurement.

Precautions in the measurement of measuring devices :

Ordinary scale : We often use ordinary scale for measuring length. It is usually a thin rectangular bar of box-wood, metal or plastic.

Precautions in measurement :

- The scale is to be placed on the straight line in such a way that the graduations of the scale be perpendicular to the straight line. By doing this the error in the reading due to thickness of the scale is avoided.
- It is better not to use any end of the scale, for that edge may be broken.
- The length of a straight line should be measured by different parts of the scale and average length should be determined.
- By doing this, the error in the reading due to irregularities in the graduations in the different parts of the scale, if any, is eliminated.
- IN.B. In making an ordinary scale wood or plastic is usually used instead of metal. With the change in atmospheric temperature metal scales changes in length.]

Measuring cylinder : A measuring cylinder is used for measuring the volume of a liquid. It is a glass cylindrical jar graduated in millilitre. The internal volumes of the jar are marked with horizontal marks, the reading starting from the bottom of the jar. The liquid whose volume is to be determined, is poured, into the jar. The volume of the liquid is obtained from the reading corresponding to the level of the liquid in the jar.

Precautions in measurement :

- During determination of volume of a liquid, it should be noted that free surface of the liquid in the jar is always either concave or convex.
- In either case (concave or convex), the reading is to be taken along a line tangential to the curved surface of the liquid. The reading is taken avoiding parallax error.

Common balance : In the laboratory we measure the mass of a body usually with a common balance. Actually we find the mass of the body by comparing its mass with some standard weights. Common balance is much more sensitive and masses of even small objects can be determined accurately with its help.

Precautions in measurement :

- Body whose mass is to be determined must be dry and at room temperature.
- The weights must be held with a pair of forceps.
- The balance beam should be on the beam support when weights are being placed on or taken away from the scale pan, otherwise the beam may topple down.

Dimensions : Dimensions of a physical quantity give the relation of its unit with the units from which it is derived.

The dimensions of a physical quantity are expressed as the powers to which the fundamental units of masses, length and time are raised to obtain the derived unit of the quantity.

Fundamental units from which the unit of a physical quantity are derived are each expressed in capital letters.

For example, the length is denoted by [L], unit of mass by [M], unit of time [T] etc. The dimensions are always written within square bracket.

w Dimensonal formula and equation : The dimensional formula of a physical quantity is an expression which gives the fundamental units on which the physical quantity depends and the nature of the dependence.

Thus, the dimensional formula of velocity is [M° L^{1} T^{-1}]. If we represent velocity by [v], then [v]=[ M°L^{1}T^{-1}] is called dimensional equation of velocity. So, when a physical quantity is equated to its dimensional formula, we get the dimensional equation of the physical quantity.

Dimensional formulae of some physical quantities :

Different types of variables and constants :

Dimensional constants : The quantities which have dimensions but of constant value are called dimensional constants.

Example : Gravitational constant, Planck’s constant.

Dimensionless constants : The constant quantities having no dimensions are called dimensionless constants.

Example : pure numbers 1,2,3 …., π, e ( = 2 .718).

Dimensional variables : The quantities which have dimensions but do not have any fixed value are called dimensional variables.

Example: volume, velocity, force etc.

Dimensionless variables : The quantities which have neither dimensions nor any constant value are called dimensionless variables.

Example : angle, specific gravity, strain etc.

Use of dimensional analysis :

The dimensional analysis have following applications :

- To check the correctness of a physical equation.
- To derive the relation between different physical quantities involved in a physical phenomena.
- To convert from one system of units to another.

Limitations of dimensional analysis :

- This method fails to determine the dimensionless constants in the formula.
- If a physical quantity depends on more than three factors, having dimensions, the formula cannot be determined.
- This method cannot be used to derive a relation if it involves trigonometric or exponential or log functions.
- This method fails to derive an exact form of a relation when it consists of more than one part on any one side.
- It gives no information whether a physical quantity is scalar or vector.
- Even when dimensions are given, the physical quantity may not be unique, as many physical quantities have the same dimensions.

Some uses of different measuring devices :

- Determination of areas of an irregularly shaped sheet of metal or paper: This is done with the help of a graph paper.
- Determination of the length of a curved line: This is done by using a thread and a linear scale.
- Determination of thickness of a sheet of a thin paper : This is indirectly done with the help of linear scale.
- Determination of volume of an irregularly shaped solid: This is done by using a measuring cylinder.
- Determination of rate of fall of water from a tap : This is done by using measuring cylinder and stopwatch.