Showing posts with label UNITS AND DIMENSIONS. Show all posts
Showing posts with label UNITS AND DIMENSIONS. Show all posts

How to Round Off Number for Physics Caliculation

The process of omitting the non significant digits and retaining only the desired number of significant digits, incorporating the required modifications to the last significant digit is called rounding off the number.

In Physics calculation is a vital part and during that we shall reduce the number to the required extend and this is called rounding off and thus we are ready to accept the error up to some extend.

Rules for rounding off numbers:

1. The preceding digit is raised by 1 if the immediate insignificant digit to the dropped is more than 5.

Ex: 4728 is rounded off to three significant figures as 4730.

2. The preceding digit is to be left unchanged if the immediate insignificant digit to be dropped is less than 5.

Ex: 472 is rounded off to three significant figures as 472

3. If the immediate insignificant digit to be dropped is 5 then there will be two different cases

a) If the preceding digit is even, its is to be unchanged and 5 is dropped.

Ex: 4.7258 is to be rounded off to two decimal places. The digit to be dropped here is 5 (along with 8) and the preceding digit 2 is even and hence to be retained as two only
4.7258=4.72

b) If the preceding digit is odd, it is to be raised by 1

Ex: 4.7158 is to be rounded off to two decimal places. As the preceding digit 1 is odd, it is to be raised by 1 as 2.

4.7158=4.72

Rules for Arithmetic Operations with significant Figures:

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

Ex: . But the result should be limited to the least number of significant digits-that is two digits only. So Final answer is 9.9.
2. In addition or substraction the final result should retain only that many decimal places as are there in the number with the least decimal places.

Ex: 2.2+4.08+3.12+6.38=15.78. Finally we should have only one decimal place and hence 15.78 is to be rounded off as 15.8.

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Errors and Significant Figures in Physics

Physics is a subject of measurements and errors do creep in while measuring physical quantities.Here we are going to learn about significant figures and how to apply them while measuring physical quantities.

Significant Figures :

A significant figure is defined as the figure, which is considered reasonably, trust worthy in number.

Eg: = 3.141592654 (upto 10 digits)
=3.14 (with 3 figures )
=3.1416 (upto 5 digits )

The significant figures indicate the extent to which the readings are reliable.

Rules for determining the number of significant figures:

1. All the non-zero digits in a given number are significant without any regard to the location of the decimal point if any.

Ex: 184,52 has five significant digits.

1845.2 or 184.52 all have the same number of significant digits that is 5.

2.All zeros occruing between two non zero digits are significant without any regard to the location of deciman point if any.

Ex: 106008 has six significant digits.

106.008 or 1.06008 has also got six sifnificant digits.

3.If the number is less than one, all the zeros to the right of the decimal point but to the first non-zero digit are not significant.

Ex: 0.000308 has 3 significant digits.

a) All zeros to the right of a demcimal point are significant if they are not followed by a non-zero digit.

Ex: 30.00 has 4 significant digits

b) All zeros to the right of the last non-zero digit after the decimal point are significant.

Ex: 0.05600 has 4 significant digits

4. All zeros to the right of the last non-zero digit in a number having no decimal point are not significant.

Ex: 2030 has 3 significant digits

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Errors and Approximation in Physics part two

This present post is in continuation with what are the errors in physics and here we are going to discuss further types of errors.

We do measure different physical quantities like velocity,acceleration,force and work done and the list goes on in physics during our experimentation.During their measurement errors do occur and we have to minimize them.

RANDOM ERRORS :

They are due to uncontrolled disturbances which influence the physical quantity and the instrument.

Ex-:The errors due to line voltage changes and backlash error.
Backlash errors are due to screw and nut.

GROSS ERRORS :

The cause for gross errors are improper recording, neglect of the sources of the error, reading the instrument incorrectly, sheer carelessness .

Ex: In a tangent galvanometer experiment, the coil is to be placed exactly in the magnetic meridian and care should be taken to see that no other magnetic material are present in the vicinity.

No correction can be applied to these gross errors.

When the errors are minimized, the accuracy increases.

The systematic errors can be estimated and observations can be corrected.

Random errors are compensating type. A physical quantity is measured number of times and these values lie on either side of mean value-with random errors. These errors are estimated by statistical methods and accuracy is achieved.

Personal errors like parallax error can be avoided by taking proper care.

The instrumental errors are avoided by calibrating the instrument with a standard value and by applying proper corrections.

True value and Errors

In the measurement of a physical quantity the arithmetic mean of all readings which is found to be very close to the most accurate reading is to be taken as True value of the quantities.

Absolute Errors :

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

Absolute error is equal to |True value - measured values|

The absolute error is always positive.

The arithmetic mean of all the absolute errors is considered as the mean absolute error or final absolute error of the value of the physical concerned and it is also positive.

Relative error:

The relative error of a measured physical quantity is the ratio of the mean absolute error to the mean value of the quantity measured.

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Errors and Approximation in Physics

Physics is the study of nature and measurement of the physical quantities is vital part of its study. During the measurements of physical quantities errors do occur and here we are dealing with different kind of errors possible.

Accuracy and precision :

The numerical values obtained on measuring physical quantities depend upon the measuring instruments, methods of measurement.

A unit of measurement of a physical quantity is the standard reference of the same physical quantity which is used for comparison of the given physical quantity.

Accuracy refers to how closely a measured value agrees with the true values.

Precision refers to what limit or resolution the given physical quantity can be measured.

Types of Errors:

Uncertainty in measurement of a physical quantity is called the error in measurement.

The difference between the measured value and true value as per standard method without mistakes is called the error.

Error = True value Measured value
Correction = -error

True value means, standard value free of mistakes.

Errors are broadly classified into 3 types :

i) Systematic errors
ii) Random errors
iii) Gross errors

SYSTEMATIC ERRORS :

The errors due to a definite cause and which follow a particular rule are called systematic errors. They always occur in one direction.

Constant error :

Systematic errors with a constant magnitude are called constant errors. The constant arised due to imperfect design, zero error in the instrument or any other such defects. These are also called instrumental errors.

Zero error :

The error due to improper designing and construction.
Ex: If a screw gauge has a zero error of -4 head scale divisions, then every reading will be 0.004cm less than the true value.

Environmental Error:

The error arise due to external conditions like changes in environment, changes in temperature, pressure, humidity etc.
Ex: Due to rise in temperature a scale gets expanded and this results in error in measuring length.

Imperfection in Experimental technique or Procedure:

The error due to experimental arrangement, procedure followed and experimental techniques called Imperfection error.
Ex: In colorimetric experiments, the loss of heat due to radiation, the effect on weighing due to buoyancy of air cannot be avoided.

Personal errors or observational errors:

These errors are entirely due to personal peculiarities like individual bias, lack of proper settings of the apparatus, carelessness in taking observations.
Ex: Parallax error

You may be interested in having a look at units and dimensions concept here below

Units and Dimensions part one and two
Uses and limitations of dimensional formulas
List of dimensional formulas part one and two


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Dimensional Formula List Two

This dimensional formula list is in continuation with that of first one . It is a representation of fundamental quantities like mass, length and time in derived quantities. It gives an idea that up to what extend the fundamental quantities are raised to derived new physical quantities.The powers to which fundamental quantities are raised are called dimensions. They help us to understand the structure of physical quantities.

Here is the remaining list of dimensional formulas. They are tabled as the name of physical quantity,its formula,dimensional formula, CGS unit and SI unit.

Please click on the image for better view.

Related posts

Units and Dimensions concepts part ONE and TWO.
Uses and limitations of dimensional formula.
Dimensional formula list one

Intrinsic and Extrinsic Semi Conducting Materials
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Dimensional Formula List One

In the previous posts we had discussed about units and dimensions concepts like the basic definitions, rules for writing units and dimensions and uses and limitations of dimensional formula.Here we are going to see the list of physical quantities, their formula,dimensional formula, CGS unit and SI unit .

Please click on the image if it is not visible clearly.



Intrinsic and Extrinsic Semi Conducting Materials

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Dimensional Formula Uses and Limitations

Dimensional Formula is the representation of fundamental quantities in a physical quantity.It gives us an idea about how the derived physical quantity is depending on fundamental quantity.For basic definitions you can go through this lesson where the present one deals with advanced concepts.

Dimensional Formula is useful for the following applications.

1.To check the correctness of the given equation. This use is based on the principle of homogeneity.

Principle of homogeneity:

It states only quantities of same dimensions can be added subtracted and equated. Hence in a Physical equation every term should have same dimensions.

2 . To convert one system of units into another system.

Eg: The numerical value of 10 joule in a new system of units in which the unit of mass is 10gm, unit of length 10cm. and unit of time 10sec. can be determined using the concept that

n1u1 = n2u2

where n1 ,n2 are the numerical values of a physical quantity and u1,u2 are the different units for same physical quantity.It means the original value of physical quantity remains same even when it is represented in different system of units.3 . To derive the equations showing the relation between different physical quantities.

Eg: When a spherical body falls through a viscous medium the upward viscous force acting on it depends upon

1. radius r of the body

2. coefficient of viscosity of the medium and

3. velocity v of the body.

We can calculate the powers of them that how do they depend on this physical quantities using the dimensional analysis as shown below.LIMITATIONS OF DIMENSIONAL SYSTEM:

1 . Dimensionless quantities cannot be determined by this method. Constant of proportionality cannot be determined by this method. They can be found either by experiment (or) by theory.

2 . This method is not applicable to trigonometric, logarthmic and exponential functions.

3.In the case of physical quantities which are dependent upon more than three physical quantities, this method will be difficult.

4 . In some cases, the constant of proportionality also posseses dimensions. In such cases we cannot use this system.

5 . If one side of equation contains addition or subtraction of physical quantities, we can not use this method.

The previous post of this site deals with velocity and acceleration concept of kinematics.

Intrinsic and Extrinsic Semi Conducting Materials

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UNITS AND DIMENSIONS CONCEPTS PART TWO

This is in continuation with UNITS AND DIMENSIONS CONCEPTS PART ONE. Going through that post fist will give you more comfort in understanding the present post.

RULES FOR WRITING UNITS:

1. Symbols for a unit named after a scientist should have a capital letter. eg:N for newton, W for watt, A for ampere.

2. Full names of the units,even when they are named after a scientist should not be written with a capital letter. Eg: newton, watt, ampere, metre.

3. Units should be written either in full or in agreed symbols only.

4. Units do not take plural form.

Eg: 10kg but not 10 kgs, 20W but not 20 Ws 2A but not 2As

5. No full stop or punctuation mark should be used within or at the end of symbols for units.

Eg: 10W but not 10W.

DIMENSIONS OF PHYSICAL QUANTITY:

1.Dimensions: Dimensions of a physical quantity are the powers to which the fundamental units are to be raised to obtain one unit of that quantity.

2.Dimensional Formula : An expression showing the powers to which the fundamental units are to be raised to obtain one unit of the derived quantity is called Dimensional formula of that quantity.

3. Dimensional Constants: The physical quantities which have dimensions and have a fixed value are called dimensional constants.

Eg: Gravitational Constant (G), Planck's Constant (h), Universal gas constant (R), Velocity of light in vacuum (c) etc.,

4. Dimensionless constants: Dimensionless quantities are those which do not have dimensions but have a fixed value.

(a): Dimensionless quantities without units.

Eg: Pure numbers, π , e, Sin θ , Cos , tan ......etc.,

(b) Dimensionless quantities with units.

Eg: Angular displacement - radian, Joule's constant- joule/calorie, etc.,

5. Dimensional variables: Dimensional variables are those physical quantities which have dimensions and do not have fixed value.

Eg: velocity, acceleration, force, work, power... etc.

6. Dimensionaless variables: Dimensionless variables are those physical quantities which do not have dimensions and do not have fixed value.,

Eg: Specific gravity, refractive index, Coefficient of friction, Poisson's Ratio etc.,

PHYSICAL QUANTITIES HAVING SAME DIMENSIONAL FORMULA:
  1. Distance, Displacement, radius light year wavelength, radius of gyration (L) .
  2. Speed, Velocity, Velocity of light .
  3. acceleration ,acceleration due to gravity, intensity of gravitational field, centripetal acceleration .
  4. Impulse, Change in momentum
  5. Force, Weight, Tension, Thrust
  6. Work, Energy, Moment of force or Torque,
  7. Moment of couple
  8. Force constant, Surface Tension, Spring constant, Energy per unit area
  9. Angular momentum, Angular impulse, Plank's constant
  10. Angular velocity, Frequency, Velocity gradient,Decay constant, rate of disintigration
  11. Stress, Pressure, Modulus of Elasticity, Energy density
  12. Latent heat, Gravitational potential
  13. Specific heat, Specific gas constant
  14. Thermal capacity, Entropy, Boltzman constant,Molar thermal capacity,
  15. wave number, Power of a lens, Rydberg constant
  16. Time, RC, L R ,
  17. Power, Rate of dissipation of energy,
  18. Intensity of sound, Intensity of radiation
  19. Expansion coefficient, Temperature coefficient of resistance
  20. Electric potential, potential difference,electromotive force
  21. Intensity of magnetic field, Intensity of magnetization
During the measurement of physical quantities different kinds of Errors occur and they shall be either eliminated or at least reduced.

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UNITS AND DIMENSIONS CONCEPTS

1. Physics is a science of measurements.

2. PHYSICAL QUANTITY: Any quantity which canbe measured directly (or) indirectly (or) in terms of which any laws of physics can be expressd is called physical quantity.

3. There are two types of physical quantities.

1) Fundamental Quantity : Physical Quantities which cannot be expressed in terms of any other physical quantites are called fundamental physical quantities.

E.g. length, mass, time, temperature etc..

2) Derived Quantity :Physical Quantities which are derived from fundamental quantities are called derived quantities.

E.g. Area, density, force etc...

UNIT OF MEASUREMENT:

1. A fixed measurement chosen as a standard of measurement to measure a physical quantity is called a Unit.

2. To measure a physical quantity means to determine the number of times its standard unit contained in that physical quantity.

3. A standard Unit is necessary for the sake of 1. accuracy,2. convenience, 3. unformity and 4. equal justice to all.

4. The standard unit chosen should have the following characteristics.

1. Consistency (or) invariability
2. Availability (or) reproducibility
3. Imperishability (Permanency)
4. Convenience and acceptability

The measure of a Physical Quantity is given by a numerical value and a unit. x= nu where x is the measure of a physical quantity, n is numerical value and u is the unit.

6. The numerical value obtained on measuring a physical quantity is inversely proportional to the magnitude of the unit chosen.

NU = CONSTANT.

Fundamental unit :The unit used to measure the fundamental quantity is called fundamental unit.

e.g., Metre for length, kilogram for mass etc..

Derived unit : The unit used to measure the derived quantity is called derived unit.

e.g., m2 for area, gm cm-3 for density etc...

FUNDAMENTAL QUANTITIES AND THEIR S.I. UNITS:

1. There are seven basic quantities and two supplementary quantities in S. I. system. The names and units with symbols are given below:

2. DEFINITIONS FOR S.I. UNITS:

1. meter: meter is 1 in 299, 792, 458th part of the distance travelled by light in vaccum in 1 second.

2. Kilogram: Kilogram is the mass of a platinum - irridium alloy cylinder proto type kept at Serves, near Paris.

3. second: One second is the time taken by 9,192, 631, 770 cycles of the radiation from the hyperfine transition in ceasium - 133 atom, when unperturbed by external fields.

4. Kelvin: This is 1/273. 16 of the temperature at the triple point of water measured on thermodynamic scale.

5. Candela: Candela is the luminous intensity in a direction normal to the surface of a block body at the temperature of freezing platinum at a pressure of 101, 325 newton per square metre.

6. ampere: ampere is the current which when flowing in each of two parallel conductors of infinite length and negligible cross-section and placed one metre apart in vaccum causes each conductor to experience a force exactly 2x10-7 newton per metre length.

7. mole: mole is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon - 12.

8. radian: radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius.

9.Steradian: The solid angle subtended at the centre of the sphere of radius 1 metre by its surface of area 1 square metre. Solid angle= normal area/r2. Total solid angle that can be formed at any point in space or at the centre of a sphere is 4 π steradian.

If you are interested in knowing about errors then here is the link for the Eerrors occur during the measurements of physical quantities.

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