## Kerala Plus One Physics Notes Chapter 2 Units and Measurement

Physical Quantities (ഭൗതിക അളവുകൾ)

The quantities which can be measured directly or indirectly are called Physical quantities.

Two types

1. Fundamental or base quantities are those which cannot be expressed in simpler forms. They are usually independent of other physical quantities.

There are seven fundamental or basic quantities. They are

- Length
- Mass
- Time
- Electric current
- Absolute temperature
- Amount of matter
- Luminous intensity

2. Derived quantities are those which can be derived from the fundamental quantities. eg., area, velocity, density, volume.

Need for measurement അളവുകളുടെ ആവശ്യം

Measurement is one of the most important part of an experiment and further research on it. When a phenomenon is observed, the information are obtained by measuring physical quantities related to it. Thus measurement is necessary in Physics.

Units

Measurement of any physical quantity is made by comparing it with a standard. Such standards of measurement are known as unit. If length of a rod is 5 m, it means that the length of rod is 5 times the standard unit ‘metre’. The I physical quantity is expressed by a number (numerical measure) and its unit.

Fundamental Unit (അടിസ്ഥാന ഏകകം)

The unit of fundamental or base quantities are called fundamental or base units. They j cannot be derived from one another, eg., metre, kilogram, second.

Derived Unit

The units of other physical quantities that can be expressed in terms of fundamental units are called derived units, eg., unit of volume is m3. Unit of velocity is m/s.

System of units (ഏകകങ്ങളുടെ വ്യൂഹം )

- cgs system. It was set up in France. It is based on centimetre, gram and second as the fundamental units of length mass and time.
- fps system. Here the unit of length is foot, mass is pound and time is second.
- mks system. In this, unit of length is metre, mass is kilogram and time is second.
- SI: The international system of units.

Earlier we had several systems. They are not common today. ‘Feet’ for length and ‘pound’ for mass are examples.

Now we follow the ‘System Internationale d’ units’, SI for short. In SI there are many base units.

Basic Physical quantity | Basic Unit | symbol |

Length | metre | m |

Mass | kilogram | kg |

Time | second | s |

Electric Current | Ampere | A |

Absolute temperature | Kelvin | K |

Amount of matter | Mole | mol |

Luminous intensity | Candela | od |

Supplementary SI units

1. Radian (rad).

It is defined as the plane angle subtended at the centre of a circle by an arc equal to the radius of the circle.

2. Steradian (sr):

It is defined as the solid angle subtended at the centre of a sphere by a surface of the sphere equal in area to that of a square, having each side equal to the radius of the sphere.

Measurement of Length

The direct methods for measurement of length use:

- a meter scale for distances from 10
^{-3}m to 102 m. - a vernier calliper for distances upto 10
^{-4 }m. - a screw gauge and a spherometer for distances upto 10
^{-5}m.

For all other distances beyond these ranges, we use indirect methods.

Measurement of Large Distances

Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. An important method in such cases is the parallax method.

Look at your finger held before your face with the left eye only. Then with your right eye only. See the difference? This is an example for parallax. The basis for this parallax is the distance between your eyes.

To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b at the same time.

We measure the angle between the two directions along which the planet is viewed at these two points. The ∠ASB represented by symbol 0 is called the parallax angle or parallactic angle. As the planet is very far away,

therefore, 0 is very small. Then we approximately take AB as an arc of length b of a circle with centre at S and the distance D as the radius AS = BS so that AB = b = D 0 where 0 is in radians.

If d is the diameter of a planet, the angle subtended by the diameter of the planet at any point on Earth is called the angular diameter of the planety. The angle a can be determined from any given location on Earth by viewing the diametrically opposite points of the planet through a telescope.

Range of lengths

1. | atomic nucleus | 1 fermi | 10^{-15}m |

2. | atomic size | 1 Angstrom | 10^{-10}m |

3. | one living cell | 1 micron | 10^{-6}m |

4. | skin | 1 mm | 10^{-3}m |

5. | iris | 1 cm | 10^{-2}m |

6. | arm length | 1 m | |

7. | Mount Everest | 8.8 kms high | |

8. | atmosphere | 200 kms | |

9. | astronomical unit | 1.496 ×10^{8 }km (distance between sun and earth) | |

10. | light year | 9.46×10^{12} km (light moves in a year) | |

11. | parsec | 3.26 light years |

(par sec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second).

Measurement of Mass

Mass is the amount of matter. It is a constant and not affected by temperature, pressure and location. SI unit is the kilogram. For smaller fractions we have gram and the

milligram. But an atom is so tiny; we have the atomic mass unit or ‘u’.

1 u = 1.66×10^{-27} kg

Masses of giant stars are measured related to the mass of sun.

Solar mass = ^{M}g. So we have stars with

masses like ^{5M}Θ , ^{20M}Θ ,^{35M}Θ , etc.

Weight :

The weight of a body is the force with which a body is pulled towards the centre of the earth. It is equal to the product of the mass (m) of the body and the acceleration due to gravity (g) of the earth on body.

Thus W = mg

As the value of ‘g’ changes from place to place, so the weight of a body is different at different places.

The SI unit of weight is newton (N).

Mass | Weight |

1. Mass is the measure of inertia | Weight is the measure of gravity |

2. It is a scalar quantity | It is a vector quantity |

3. It is a constant quantity. | It varies from place to place |

4. It cannot be zero for a body | Weight of a body is zero at the centre of the earth. |

5. It is not affected by the presence of other bodies. | It is affected by the presence of other bodies. |

Measurement of Time

There are various methods to measure time.

Atomic clock is based on the periodic vibrations produced in a cesium atom, also called cesium clock. It has a very low error.

Quartz clocks are also widely used. It works due to a phenomenon called peizo electric effect.

Accuracy, Precision of Instruments and Errors in Measurement

- Accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.
- Precision tells us to how much the measured values close to each other. For every instrument, there is a minimum value that can be measured accurately. This is called
- least count of that instrument. It is 0.1 cm for an ordinary scale, 0.01 cm for an ordinary vernier callipers and 0.001 cm for an ordinary screw gauge.

Error

The difference in the true value and the measured value of a quantity is called error of measurement.

Error = True value – Measured value

In general, the errors in measurement can be broadly classified as (a) systematic errors and (b) random errors.

Systematic errors

The systematic errors are those errors that tend to be in one direction, either positive or negative.

Sources of systematic errors

- Instrumental errors
- Imperfection in experimental technique or procedure
- Personal errors

1. Instrumental errors

Instrumental error arise from the errors due to imperfect design or calibration of the measuring instrument, eg., in vernier callipers, the zero mark of vernier scale may not coincide with the zero mark of the main scale.

2. Imperfection in experimental technique or procedure

To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature.

Other external conditions (such as changes in temperature, humidity, velocity, etc) during the experiment may affect the measurement.

3. Personal errors

Personal error arise due to an individual’s bias, lack of proper setting of the apparatus or individual carelessness etc.

Random errors

Random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions

(eg., unpredictable fluctuations in temperature, voltage supply, etc.).

Least count error

The smallest value that can be measured by the measuring instrument is called its least count. The least count error is the error associated with the resolution of the instrument.

By using instruments of higher precision, improving experimental technique etc, we can reduce least count error.

Gross error

These errors are due to the carelessness of the observer.

Absolute error, Relative error, Percentage error

1. Absolute error.

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

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

a_{1 ,}a_{2,}a_{3………}a_{n .}The arithmetic mean of these values is

Usually the arithmetic mean am is taken as the best or true value of the quantity. So, absolute error in the individual measured values will be

2. Mean absolute error.

The arithmetic mean of the positive magnitudes of all the absolute errors is called mean absolute error. It is given by

Thus the final result of measurement can be written as a = a±Δa_{mean}

3. Relative error.

The ratio of the mean ab-solute error to the true value of the measured quantity is called relative error.

4. Percentage error.

The relative error expressed in percent is called percentage error.

** **Combination of Errors

a. Error of a sum or difference

If a quantity Z is expressed as the sum or difference of two quantities A and B

(ie., if Z = A + B or Z = A – B), then maximum value of error ΔZ = ΔA + ΔB.

Rule: when two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

b. Error of a product or a quotient

If a quantity Z be expressed as product or a quotient of quantities A and B, then the maximum fractional error in Z is given by

Rule: when two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

c. Error in case of a measured quantity raised to a power

If etc., then maximum fractional error in Z is given by

Rule :The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

Question 1.

The length of a rod as measured in an experiment was found to be 3.52, 3.48, 3.54, 3.50, 3.49. Find the average length, the absolute error In each observation and the percentage error.

Significant Figures

These are the digits that matter in a number or observation. The rules are:

- All non-zero digits are significant. e.g., 16.47 has four significant figures.
- All zeroes between two non-zero digits are significant.

e.g., 100.07 has five significant figures. - In a number without a decimal point, the ending zeroes are not significant.

e.g.,3200 has 2 significant figures. - If number is less than one, the zeroes on the right of the decimal point and to the left of the first non-zero digit are not significant.

e.g.,0.005704 has 4 significant figures. - In a number with decimal point, the ending zeroes are significant.

e.g., 4.700 has 4 significant figures and 0.075000 has 5 significant figures (the two zeroes left to 7 are not significant). - Change of units does not change the number of significant figures in a measurement.

e.g., 16.5 cm, 0.165 m and 0.000165 km all have three significant figures.

123 m = 12300 cm = 123000 mm all have three significant figures - In exponential notation, the numerical portion gives the number of significant figures.

e,g., 1.35 × 10^{8}has three significant figures..

1.35 ×10^{6}has three significant figures.

Question 2.

State the number of significant figures in the following:

(i) 2.000 m

(ii) 6200kg

(iii) 070 cm

Sol:

i. Four: 2, 0, 0, 0

ii. Four: 6, 2, 0, 0

iii. Two: 0, 7

Rules of Arithmetic operations with Significant figures

- In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.

e.g., Multiply 3.8 and 0.125

We have 3.8 × 0.125 = 0.475

The least number of significant figure is 2.

∴ 3.8 ×0.125 = 0.475 = 0.48 - In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

e.g., Add 16.25 kg, 24.8 kg and 9.443 kg. 16.25 + 24.8 + 9.443 = 50.493 kg

The least number of significant figure is 3.

∴ 16.25 + 24.8 + 9.443 = 50.493 = 50.5 kg

Rounding off uncertain digits

Rule 1: The preceding digit is raised by 1 if the digit to be dropped is more than 5.

7.536 is rounded off to 7.54 as the digit to be dropped (6) is greater than 5.

Rule 2: The preceding digit is not changed if the digit to be dropped is less than 5. 3.251 becomes 3.25. Here the digit to be dropped (1) is less than 5.

Rule 3: When the digit preceding the digit to be dropped is ‘even’ and the digit to be dropped is 5, then there is no change for preceding digit. 8.625 becomes 8.62. Since the digit preceding the digit to be dropped (5) is 2 which is even.

Rule 4: When the digit preceding the digit to be dropped is ‘odd’ and the digit to be dropped is 5, then 1 is added to the preceding digit. 9.535 becomes 9.54. Since the digit preceding the digit to be dropped (5) is 3 which is odd.

Dimensional Formula and Dimensional Equation** **Dimensions

All physical quantities can be represented in terms of fundamental quantities; length, mass, time, etc.

e.g., We know, Area = length x breadth. Here length and breadth are measured using the unit of length (L).

Thus, unit of area = L x L = L

^{2}.

Since mass M and time T are not needed for measurement of area, unit of area can also be represented as M°L

^{2}T°. Here the powers 0, 2, 0 of the fundamental units are called the dimension of 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.

Examples:

Volume | [M°L^{3}T°] |

Density | [ML^{3}T°] |

Displacement (s) | [M°LT°] |

Velocity (v) | [M°LT^{-1}] |

Area, A | [M°L^{2}T°] |

Force, F or Weight, W | [MLT^{2}] |

Note: Quantities like number, angle and trigonometric ratios are dimensionless.

Question 3.

Name the physical quantities whose dimensional formulae are as follows:

(i) [ML^{2}T^{-2}]

(ii) [ML^{2}T^{-2}]

(iii) [ML^{-1}T^{-2}]

Answer:

Dimensional Constants:

Constant which possess dimensions are called dimensional constants. e.g., Planck’s constant, universal gravitational constant.

Dimensional Variables:

Are those physical dimensions but do .eg., velocity, acceleration, force etc.

Dimensional Quantities:

There are certain quanties which do not possess dimensions. They are called dimensionless quantities. e.g., Strain, angle, specific gravity etc.

Dimensional Analysis and its Applications** **According to this, a physical equation will be dimensionally correct if the dimensions of all the terms on both sides of the equation are same.

The important uses of dimensional analysis are:

- To clock the correctioness of an equation
- To derive a correct relationship between different physical quantities.
- To convert one system of units into another.

1. Checking the correctness of an equation:** **For this we use the principle of homogeneity of dimensions.

Question 4.

Let us check the dimensional accuracy of the equation of motion.

Answer:

For this purpose we make use of the principle of homogeneity of dimensions. If the dimensions of all the terms on the two sides of the equation are same, then the equation is dimensionally correct.

S = ut + 1/2 at^{2 }Dimensions of different terms are

[s] = [L]

[ut] = [LT]

[ut]=[LT^{-1}] [T] = [L]

[1/2 at^{2}] = [LT^{-2}] [T^{2}] = [L]

As ail the terms on both sides of the equations have the same dimensions, so the given equation is dimensionally correct.

2. Deriving the correct relationship between different physical quantities:

The principle of homogeneity of dimensions also helps to derive a relationship between the different physical quantities involved.

Question 5.

Derive an expression for the centripetal force F acting on a particle of mass m moving with velocity v in a circle of radius r.

Sol:

Let F α m^{a}v^{b}r^{c} or F = Km^{a}v^{b}r^{c} …(1)

where K is a dimensionless constant. Writing the dimensions of various quantities in equation (1), we get

Comparing the dimensions of similar quantities on both sides, we get

a=1,b + c= 1 ore = 1 -b = 1 -2= -1

2 = -b or b = 2

From equation (1), we get

This is the required expression for the centripetal force.

3. To convert one system of units into another.

It is based on the fact that magnitude of a physical quantity remains the same, whatever be the system of its measurement.

Question 6.

Convert one Joule Into erg.

Answer:

Joule Is SI unit of energy and erg is the CGS unit of energy. Dimensional formula of energy Is ML^{2}T^{-2}

∴ a = 1, b = 2, c = -2

SI | CGS |

M_{1 }= 1 kg = 1000 g | _{M2}=1g |

L_{1}=1m = 100cm | L_{2} = 1 cm |

T_{1 =} 1s | T_{2}= 1s |

n_{1} a 1 (joule) | n_{2}=? (erg) |

* *Limitation of Dimensional Analysis:

The method of dimensional analysis has the following limitations:

- The value of dimensionless constants cannot be determined by this method.
- This method cannot be applied to equations involving exponential and

trigonometric functions. - It cannot be applied to an equation involving more than three physical quantities.
- It can check only whether a physical relation is dimensionally correct or not. It cannot tell whether the relation is absolutely correct or not.

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