Physical World and Measurement : Units and Measurements

Chapter–2: Units and Measurements

Chapter–2: Units and Measurements : Need for measurement: Units of measurement; systems of units; SI units,
fundamental and derived units. significant figures. Dimensions of physical quantities, dimensional analysis and its applications.

12 PHYSICALQUANTITIES:-

The quantities which obey law of physics are called physical quantity. Everything in this nature that we see or measure is physical quantity. To study physical quantity we need number and units.

⇒Physical quantity =number + unit

Number: –

How many times a physical quantity is taken is called number.

Unit:-

The standard by which we measure physical quantity is called unit.

13 MEASUREMENTS

Hence measurement of a physical quantity =numerical value of the physical quantity× size of the unit

For example, let length of a rod = 5m=500cm.

Here smaller the size of the unit larger is the numerical value, thus numerical value(n) is inversely proportional to size(u) of the unit.

Also we may write

A good unit will have the following characteristics.

It should be (a) well defined (b) easily accessible (c) invariable (d) easily reproducible

Question:1The SI unit of force is Newton such that 1N = 1kg. In C.G.S. system, force is expressed in Dyne.

How many dyne of force is equivalent to a force of 5 N?

Solution: Let 1 N = n dynes



 5 N = 5 × dyne.

Units can be divided into two parts

(a) Fundamental units. (b) Derived units

(a) Fundamental units:-

The units which can neither be derived from another unit, nor can they be further resolved.

E.g. Length, mass, time is the fundamental units.

(b) Derive Units:-

The units which can be expressed in the form of fundamental units are called derived units.

Therefore, units of speed

Rules for writing units

 The symbol of a unit, which is not a name of a person, is written in small letter.

E.g.: metre (m) kilogram (kg)

 The symbol of a unit, which is given on the name of a person, is written with a capital initial letter.

E.g.: Newton (N), Kelvin (K)

 Full name of a unit, even if it is named after a person is written with a lower initial letter.

E.g.: Newton, Kelvin.

 A compound unit formed by multiplication of two or more units is written after putting a dot or leaving a space between the two symbols.

E.g.: Newton metre – N.m or N m.

 A unit in its short form is never written in plural.

i.e.5 Newton may be written as 5N not 5Ns

14 SYSTEMS of units

The F.p.s. system

In this system of unit’s fps represents foot, pound, second for the measurement of length mass time respectively.

It was the British system of measurement.

The c.g.s system

In this system of unit’s c.g.s represent centimeter, gram, second for the measurement of length mass time respectively.

The m.k.s system

In this system of unit’s m.k.s represents metre, kilogram, second for the measurement of length mass time respectively.

Now a day’s only c.g.s. and m.k.s. system of units are used.

The cgs system is used for small quantities.

International system of units (SI)

The S.I system of units was developed by General Conference of Weight and Measurement in 1971.

It consists of 7 fundamental units and 2 supplementary units.

Aberration in power of ten

15. Characteristics of a system of unit/properties of SI units:-

1.It is well defined.

2. It is of suitable size i.e. neither too large nor too small in comparison to the quantity to be measured.

3.It is easily reproducible at all places.

4.It does not change with time and from place to place.

5.It does not change with change in its physical condition, such as temperature, pressure, etc.

6.It is accessible easily.

7.It is internationally accepted.

16 Advantages of the SI system of the unit over other system of unit

Si is coherent system of the unit:-

all derived unit can be obtained by simple multiplication or division of the units and the numbers

Si is rational:-

it uses only one unit for one physical quantity

e,g all form of energy are measured in joule.

It is metric system :-

multiple and submultiples of SI unit can be expressed in power of 10

It is absolute system: –

it does not use gravitational units. The use of g is not required.

It is internationally accepted.

17 INTRODUCTIONS TO PHYSICAL QUANTITIES AND THEIR S.I. UNITS

MASS

The quantity of matter contained in a body is called mass. It can never be zero for a body.

The SI unit of mass is kilogram

One kilogram is defined as the mass of one cubic decimeter of water at 4⁰C (the temperatures of water at which its density is maximum or

One kilogram is defined as the mass of a platinum-iridium cylinder placed at international Bureau of weight and measurement near Paris, France.

Measurement of Mass

There are two types of mass of a body.

I Inertial mass

The mass under the effect of an external force rather than gravity is called inertial mass.

It is measured by spring balance

ii. Gravitational mass

When the body is under the effect of gravity, than the measured mass is called gravitational mass,

it is measured by a physical balance.

Both inertial mass and gravitational mass are equal.

Difference between mass and weight

Mass

The quantity of matter contained in a body is called mass.
Mass is a measure of inertia
It is scalar quantity
It cannot be zero
It is essential property of a material body
Not affected by presence of other body
Its unit are g, kg,

Weight

The force with which a body is pulled toward center of earth is called weight.
Weight is a measure of gravity.
it is a vector quantity
it is zero at Centre of earth
It is not essential property of a material body
affected by presence of the other body
Its unit are dyne, Newton etc.

SOME IMPORTANT PARACTICAL UNITS OF MASS

1 atomic mass unit =1.66×10^-27kg1 tone or 1 metric ton 1000kg

1 quintal =100kg1 slug = 14.57kg1 pound =1lb=0.4536kg

1 Chandra Shekher limit= 1CSL=1.4 time the mass of the sun.(largest unit of mass)

(b) LENGTH

Length is defined as the separation between two points in free space.

It is measure in metre.

One metre is defined as the path followed by light in vacuum in 1/299,792,458 of a second

Measurement of length

Now a day we can measure length from 10^-16 m (diameter of electron) to 10²⁶(size of universe)

Measurement of length is done by two methods

1 Direct method 2 indirect method

I. Direct method

In this method length is measured by instruments i.e. by metre scale, a Vernier caliper, a screw gauge, or a speedometer.

The minimum distance measured by these instruments is called least count of the instrument.

ii. Indirect method used in length measurement:-

It is used to measure height of a tower, poles, mountains, distance of moons and other celestial objects from earth.

(a) Measurement of the large distance

Parallax method:-

This method is used for measuring distance of planet and stars distance less than 100 light year.

Parallax means the change in position of an object w.r.t. background when we shift our eyes sidewise.

Let a star which is fixed at O is seen from earth from two different points A and B having equal distance R from the star.

Let the distance between A and B is L which may be taken as the arc because L << R.

Then the paratactic angle θ can be given as

(1)

The angle θ can only be measured if the position of O is measured at the same time, which is not possible.

Than another star F is considered whose distance does not change with time such that.

θ = θ₁+ θ₂

Than from (1)

Hence distance from stars can be measured.

Triangular method for the measurement of an accessible object

Suppose is the height of a tower or a tree which is to be measured.

Then this method is used. Let A is the point of observation having distance from the base of the tree B. now place a sextant at c and measure the angle of elevation θ.

Now from right angle triangle ABC, we have

Or height =

By knowing the distance, the height can be determined.

Triangular method for the height of an inaccessible object

Let AB is the height of the object which is to be measured.

By using a sextant we measure first angle C then angle D As shown in fig.

Now from right angle triangle ABC

And in triangle ABD

Knowing d, the height h can be determined.

Example2: A man wishes to estimate the distance of a nearby tower from him. He stands at a point A in front of the tower C and spots a very distant object O in line with AC.

He then walks perpendicular to AC up to B, a distance of 100 m, and looks at O and C again.

Since O is very distant, the direction BO is practically the same as AO; but he finds the line of sight of C shifted from the original line of sight by an angle θ = 40⁰(θ is known as parallax) estimate the distance of the tower C from his original position A.

Answer We have, parallax angle

From Fig. , and

Determination of the distance of a far away star by intensity method

This method based on the inverse square law of intensity. I.e. the intensity of the illumination at a point is inversely proportional to the square of distance from the source of light.

Suppose I₁ is the intensity of the far away star and I₂ is the intensity of the nearby star taken on a photographic plate.

Let r₁ and r₂ is the respective distances of stars.

Than from inverse square law of intensity

Knowing the distance r₂ of the nearby star, the distance of the far away star can be calculated.

This method is used to measure the distance of the stars more than 100 light years.

Q. what do you mean by inferior and superior planet?

Inferior planet

The planets which are closer to the sun then earth are called inferior planet (mercury and Venus)

Superior planet:

The planets which are closer to the earth then sun are called superior planet (Jupiter, Saturn, Uranus, Neptune, and Pluto)

Measurement of the diameter of the moon

Suppose AB=D is the diameter of the moon which is to be measured by an observer O on the earth. As shown in the fig.

Now from triangle OAB

Hence

Knowing the value of S and the diameter D can be measured

Example: 3the moon is observed from two diametrically opposite points A and B on Earth.

The angle θ subtended at the moon by the two directions of observation is.

Given the diameter of the Earth to be about, compute the distance of the moon from the Earth.

Answer We have

Since Also

We have the earth-moon distance,

Example: 4 The Sun is angular diameter is measured to be 1920′′.The distance D of the Sun from the Earth is m. What is the diameter of the Sun?

Answer Sun is angular diameter

Sun is diameter

Reflection or echo method

This method is based on the speed of the sound. To find the distance of a hill a gun is fired toward the hill, the time interval between the instant of the firing and instant of the hearing is noted.

During this time interval the sound travels 2S distance with v velocity.

So the distance traveled by the light may be given as2S=v × t

Or S=

Laser method

The word laser means light amplification by the stimulated emission of the radiations.

A laser beam is sent toward moon and its reflected pulse is received. If t is time elapsed between the instants of light sent and light received back, then the distance of the moon from the earth is given by

Where c=3 10⁸m/s is the velocity of the light traveled.

RADAR method

The word RADAR means radio detection and ranging. Radar is used to measure the distance of a nearby planet. A radio wave is sent from a transmitter and after reflection from the planet received by a detector. Then the distance travelled by the radio waves can be given as

This method is also used to measure the height and the distance of an aeroplane.

SONAR method

The word sonar means sound navigation and ranging. On sonar, ultrasonic wave of frequency greater than 20,000Hz are transmitted through the ocean. They are reflected by the rock or the submerged rocks and received by the receiver.

Then the distance travelled by the sound waves toward rock can be given as

(ii)Measurement of the small distance

Small distances such as size of atom, molecules are measured by electron microscope, Avogadro method, Rutherford α particle scattering method etc.

Measurement of the small distance (Size of molecule of oleic acid)

Oleic acid is a soapy liquid with large molecular size. Dissolve 1 cm³ of oleic acid in 20 cm³ of alcohol and then re-dissolve 1cm³ of this solution in 20 cm³ of alcohol. Then the concentration of oleic acid is 1/400 cm³ of alcohols.

We than determine approximate volume of each drop (V cm³). Now pour n drops of this solution on surface of water.

We stretch the film carefully, as alcohol is evaporate, a very thin film of left on water surface. We measure the area A of film using graph paper.

Volume of the n drops of the solution

= cm³.

Amount of oleic acid in the solution

= cm³.

Thickness of oil film

t=cm.

The value of t is found to be of order of 10^-9m.

SOME IMPORTANT PARACTICAL UNITS OF DISTANCE

(a) For large distance

1. Astronomical Unit (AU):-

The average distance from center of earth to center of moon

1 AU=1.496×10¹¹ m≃1.5×10¹¹m

2. Light year (ly):-

The distance travelled by light in vacuum in one year

1 ly=3×10⁸×365×24×60×60=9.46×10¹⁵ m

3 parsec:-

It is the distance at which an arc of length equal to one astronomical unit subtends an angle of one second.

Relation between AU, ly, Parsec

1 ly=6.3×10⁴ AU

1 Parsec =3.28 ly

1 Parsec=2.07×10⁵ AU

1 barn =10^-28m²

1 acre= 4047m²

1 hectare =10⁴m²

(c) TIME

Time is defined as the duration between two events. The SI unit of time is second.

One second is defined as the duration of 9,192,631,770 vibrations between two hyperfine levels of cesium-133 atom in ground states.

Measurement of Time:-

Time is measured by solar clock, quartz crystal clock, atomic clock, decay of elementary particle, age of rocks, earth etc

Some practical units for the measurement of the time

Solar day

The time taken by the earth to compete one rotation about its own axis w.r.t the sun is called a solar day.

1 solar day =24hour= 86400 s

Sedrial day

The time taken by the earth to compete one rotation about its own axis w.r.t a distant star is called a Sedrial day.

Solar year

The time taken by the earth to compete one revolution around the sun in its orbit is called a solar year.

1 solar year=365.25 average solar day

Tropical year:

The year in which there is total eclipse is called one tropical year.

Leap year

The year which is total divisible by number 4 is called leap year, in this year the month of February have 29 days.

Lunar month

The time taken by the moon to complete one revolution around the earth is called one lunar month.

1lunar month =27.3 days

The smallest unit of time is shake 1 shake=10^-8s

Now a days we can measure time from 10^-24 to 10¹⁷second (from life time of unstable particles in nucleus to age of universe)

CURRENT:

Rate of flow of charge in a conductor is called current. It is measured in Ampere.

One Ampere current is defined as the current between two parallel straight conductors of infinite length, negligible area of cross-section and placed one metre distance apart in vacuum would produce a repulsive force equal to 2×10^-7. N.

TEMPRATURE:

Temperature is defined as the degree of hotness or coldness of a body. It is measured in Kelvin

One Kelvin is defined as the 1/273.16 the fraction of temperature at the triple point (the point at which ice water and water vapor coexist) of the water

LUMIONUS INTENSITY:

Luminous intensity is defined as the amount of light emitted per second by a source. It is measure in candela.

One candela is defined as the light emitted perpendicularly by 1/60,000 sq. m area of a black body at freezing point of platinum at 2042k temperature and 101,325N/m² pressure.

(g) QUANTITY OF MATTER

Amount of a substance contain in a body. It is measure in mole

One mole is defined as amount of substance which contains same number of atoms as there are atoms in 0.012kg of carbon C-12.

SUPPLYMEANTRY UNIT

1 PLANE ANGLE

Plane angle is defined as the angle between two planes. It is measured in radian.

One radian is defined as the angle subtended at the center of the circle by an arc of length equal to radius of the circle.

1 radian=57.7⁰

2 SOLID ANGLE

Solid angle is defined as the angle at the Centre of sphere

Physical World and Measurement It is measure in steradian.

One steradian is defined as the angle subtended at the center of sphere by its surface whose area is equal to the square of radius of the sphere.

Example: 5 Calculate the angle of (a) (b) 1′ and (c) 1″ in radians.

Use rad, and 1′ = 60 ″

Answer (a) we have

(b)

(c)

DIMENSIONAL ANALYSIS

18 DIMENSIONS

Dimensions of a physical quantity may be defined as the powers to which the fundamental units be raised in order to represent that quantity.

For example: we know, Area = length x breadth.

Thus unit of area = L L = L².

Since unit of mass M and time T are not being used for the measurement of area, the unit of area can be represented by M⁰L²T⁰.

Powers 0, 2, 0 of fundamental units are called the dimensions of area in mass, length and time respectively.

** Dimensional equation

The equation, which indicates the units of a physical quantity in terms of the fundamental units, is called dimensional equation.

Eg: Dimensional equation of velocity is [V] = M⁰LT^–1.

1) Area = length x breadth = L x L = L² = M⁰L²T⁰

2) Volume = L³ = M⁰L³T⁰

19 DIMENTIONAL FORMULAS AND SI UNIT OF SOME PHYSICAL QUANTITY

20 Different types of variables and constants:

Dimensional variable

The physical quantity having dimensional formula and variable value is called dimensional variable. E.g. area, volume, velocity, force

Dimensional constant

The physical quantity having dimensional formula and constant value is called dimensional constant. Gravitational constant, Planck’s constant

Dimensionless variable

The physical quantity having no dimensional formula but have variable value is called Dimensionless variable. E.g. angle, strain

Dimensionless constant

The physical quantity having no dimensional formula and have constant value is called Dimensionless constant. Π, e, etc.

21 Principle of homogeneity

An equation representing a physical quantity will be correct if the dimensions of each term on both sides of the equation are the same. This is called the principle of homogeneity of dimensions.

22 Uses of dimensional analysis

To check the correctness of an equation.
To convert one system of unit into another system
To derive the correct relationship between physical quantities.

To check the correctness of an equation.

An equation is correct only if the dimensions of each term on either side of the equation are equal.

Example: 6 Check the accuracy of the equation S=ut+at²

Ans given formula is S=ut+at²

Taking dimensions on both sides,

L = L +L

i.e. L = L + L

According to principle of homogeneity, the equation is dimensionally correct.

Example:7 Let us consider an equation . Check whether this equation is dimensionally correct.

Answer the dimensions of LHS are

The dimensions of LHS and RHS are the same and hence the equation is dimensionally correct

Question: 8 The equation of state of some gases can be expressed as.

Here, P is the pressure, V the volume, T the absolute temperature, and a, b, R are constants. Find the dimensions of a.

Solution: Dimensions of will be same as dimensions of pressure

= ML^–1T^–²

so [a]= ML⁵T^–2

To convert one system of unit into another system

This is based on the principal that the magnitude of the physical quantity remains same, whatever is its unit.

Suppose a physical quantity Q having u₁ and u₂ units with corresponding numbers n₁ and n₂.

Then Q= n₁u₁=n₂u₂

Again suppose that M₁L₁T₁ are the fundamental units in one system and M₂L₂T₂ are the fundamental units of mass length and time in second system.

If the dimension formula of the physical quantity is

Then

And

So =

This equation may be used to find the numerical value of the second system of unit.

Q.9 convert one joule of energy into erg?

Here joule is the SI unit of the energy and erg is the cgs unit of the energy. The dimensional formula of energy is ML^-2T^-2.

o r

=1×10³×10⁴=10⁷

To derive the correct relationship between physical quantities.

Dimensions can be used to derive the relation between physical quantities.

Question To derive an expression for period of oscillation of a simple pendulum.

The period of oscillation (T) of a simple pendulum may depend on(1) Length of the pendulum (L);(2) Mass of bob (m) and (3) the acceleration due to gravity (g).

Then T =

Taking dimensions on both sides =

Equating dimensions of M, L and T,

x + z = 0

y = 0

–2z = 1

Hence z =-and x=

By putting the values we get T=

Or T=

The value of constant k cannot be found by dimensional method.

The value of k is found to be 2π using some other methods.

Question:10 In a new system of units, unit of mass is taken as 50 kg, unit of length is taken as 100 m and unit of time is 1 minute. What will be the weight of a body in this system, if in SI system, its weight is 10 N.

Solution: Let the weight of the body in new system is X units

 10 N = X units

Let and be the symbols for mass, length and time in the two system respectively,

then



10 N = 7.2 units

23Limitations of dimensional analysis.

 This method gives us no information about dimensionless constants.

 We cannot use this method if the physical quantity depends on more than three other physical quantities.

 This method cannot be used if the left hand side of the equation contains more than one term.

 Often it is difficult to guess the parameters on which the physical quantity depends.

1(d) Significant figures AND error

24 Significant figures:

Significant figures give the number of meaningful digits in a number.

The significant figures are the number of digits used to express the measurement of the physical quantity such that the last digit in it is doubtful and the rest all digits are accurate.

The number of significant figures depends on the accuracy of the instrument. More the number of significant figures in a measurement, more accurate the measurement is.

25 Rules to determine the significant figures:

(1) All zeros in between the numerals 1 to 9 are counted.

(2) In a measurement involving decimal, the position of decimal is disregarded.

(3) All zeros after the last numeral are counted.

(4) The zeros preceding the first numeral are not counted.

(5)All the zeros to the right of the last non-zero digit (trailing zeros) in a number without a decimal point are not significant, unless they come from experiment.

Thus 123 m = 12300 cm = 123000 mm has three significant figures. The trailing zeros are not significant. But if these are obtained from a measurement, they are significant.

26 Rules for rounding off to the required number of significant figures:

(1) If the digit to be dropped is less than 5, the digit immediately preceding it remains unchanged.

(2) If the digit to be dropped is more than 5, the digit immediately preceding it is increased by 1.

(3) If the digit to be dropped is 5, then the preceding digit is made even by

(a) Increasing it by 1 if it is odd,

(b) Keeping it unchanged if it is even.

27 Significant figures in calculations

(1) Significant figures in multiplication and division.

The result of multiplying or diving two or more numbers can have no more significant figures than those present in the number having the least significant figures.

Eg:(1) [sig fig 4] [sig fig 5]

Than

(2) [sig fig 4]

Then

(2) Significant figures in addition and subtraction.

In adding or subtracting, the least significant digit of the sum or difference occupies the same relative position as the least significant digit of the quantities being added or subtracted. Here number of significant figures is not important; position is important.

Eg:(1)204.9[9 is least sig. digit. Position – 1st decimal place]

2.10[0 is least digit. Position 2nd decimal place]

+ 0.319[9 is least sig. Digit. Position 3rd decimal place.]

= 207.319 = 207.3

In sum, the least sig. Fig should come in the first decimal place.

Eg:(2)If a = 10.43 and b=2.8612

Then a – b= 10.43 – 2.8612

= 7.5688 = 7.57.

Example11. 5.74 g of a substance occupies 1.2 cm³. Express its density by keeping the significant figures in view.

Answers There are 3 significant figures in the measured mass whereas there are only 2 significant figures in the measured volume. Hence the density should be expressed to only 2 significant figures.

Example12.Each side of a cube is measured to be 7.203 m. What are the total surface area and the volume of the cube to appropriate significant figures?

Answer The number of significant figures in the measured length is 4.The calculated area and the volume should therefore be rounded off to 4 significant figures.

Surface area of the cube =

Volume of the cube =

ACCURACY AND PRECISION OF INSTRUMENTS

Accuracy of an instrument represents the closeness of the measured value of actual value. Precision of an instrument represents the resolution of the instrument. It depends on least count.

Physical World and Measurement

Least count of an instrument is the least measurement, which can be made accurately with that instrument.

Least count of an ordinary metre scale is 0.1 cm, 0.01 cm is the least count of Vernier calipers and 0.001 cm is that for screw gauge.

For example a physical quantity is measured from two instruments A and B. The reading of A is 2.54cm (say) and that of B is 2.516 cm.

The actual value is 2.53 cm. The first reading is closed to actual value, it has more accuracy. The second reading is less accurate, but the instrument B has greater resolution as it can measure up to 3 decimal places.

28 Errors in Measurements

Difference between the value obtained in a measurement and the true value of the quantity is called error in that measurement.

The following are the commonly occurring errors:

Constant error:

If the error in a series of readings taken with an instrument is same, the error is said to be constant error.

Systematic errors:

Errors, which are due to known causes and act according to a definite law are called systematic errors.

Instrumental errors:

These errors are due to the defect of the instrument.

Eg: 1.Zero error in screw gauge and Vernier.

2.Faulty calibration of thermometer, metre scale etc.

(b) Personal error:

This is due to the mode of observation of the person taking the reading. E.g.: Parallax error.

Error due to imperfection:

This is due to imperfection of the experimental setup.

Eg: Whatever precautions are taken, heat is always lost from a calorimeter due to radiation.

Error due to external causes:

These are errors caused due to change in external conditions like temp, pressure, humidity etc.

Eg: With increase in temperature, a metal tape will expand. Any length measured using this tape will not give corrects reading.

Random Error:

The errors, which occur irregularly and at random in magnitude and direction, are called random errors.

These errors are not due to any definite cause and so they are also called accidental errors.

Random errors can be minimized by taking several measurements and then finding the arithmetic mean.

The mean is taken as the true value of the measured quantity.

Let a quantity measured n times give values , then the possible value a of the quantity is

The arithmetic mean is taken as the true value of the quantity.

Gross Errors:

The errors caused by the carelessness of the person are called gross errors. It may be due to

(1) Improper adjustment of apparatus

(2) Mistakes while taking and recording readings etc.

(5) Absolute Errors:

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

If a₁, a₂,…..,a_n are the measured values and is the true value,

Then absolute error,

Δand so on.

Mean absolute error:

The arithmetic mean of the absolute error of the different measurements taken is called mean absolute error.

If Δ, Δ,……. Δ are the absolute errors in the measurements a₁, a₂, a₃,…a_n, then

7) Relative and percentage errors.

The ratio of the mean absolute error to the true value of the measured quantity is called relative error.

And percentage

Example13.We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80 s. calculate the absolute errors, relative error or percentage error.

Answer The mean period of oscillation of the pendulum

The errors in the measurements are

The arithmetic mean of all the absolute errors (for arithmetic mean, we take only the magnitudes) is

That means, the period of oscillation of the simple pendulum is (2.62 ± 0.11) s

i.e. it lies between (2.62 + 0.11) s and (2.62 – 0.11) s or between 2.73 s and 2.51 s.

As the arithmetic Mean of all the absolute errors is 0.11 s,

there is already an error in the tenth of a second. Hence there is no point in giving the period to a hundredth.

A more correct way will be to write T = 2.6 ± 0.1 s

The relative error or the percentage error is

30 Combination of errors

(1) Error in a sum or difference: —

Let two quantities A and B have errors Δ A and Δ B respectively in their measured values. We have to calculate the error Δ Z in their sum Z = A + B.

We will get the same result even if we take the difference.

 When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the quantities.

Example14 The temperatures of two bodies measured by a thermometer are

Calculate the temperature difference and the error there in.

Answer

(2) Error in a product or quotient

Let A and B be two quantities and Z be their product.

i.e. Z = AB. If A and B are the measured values of A and B, then

Z = A)( B)=ABABABAB

Dividing LHS by Z and RHS by AB,

ie, being small.

So neglecting

Hence the maximum possible relative error or fractional error in Z is

The result is true for division also.

Therefore, when two quantities are multiplied or divided, the relative error of the result is equal to the sum of relative errors of the quantities.

Example15.the resistance where V = (100 ± 5) V and I = (10 ± 2) A.

Find the percentage error in R.

Answer The percentage error in V is 5% and in I it is 2%.

The total error in R would therefore be 5% + 2% = 7%.

Example16. Two resistors of resistances = 100 ±3 ohm and = 200 ± 4 ohm are connected in series.

Find the equivalent resistance of the series combination.

Use the relation,

Answer the equivalent resistance of series combination

(3) Error when a quantity is raised to a power.

Consider a quantity

Taking logarithm on both sides, log X = log xⁿ = n log x.

Differentiating, we get=

Thus if a quantity has to be raised to a power n, then the relative error of the result is n times the relative error of that quantity.

Example17. The period of oscillation of a simple pendulum is Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution.

What is the accuracy in the determination of g?

Answer

Here,

Example18. A physical quantity P is related to four observables a, b, c and d as follows: .

The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively,

What is the percentage error in the quantity P ?

Ans

To download class 11 physics unit 1 Physical world and measurement notes click on the link given below

unit 1 Physical world and measurement