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Physics And Measurement

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Physics and Measurement PDF Notes, Important Questions and Formulas

PHYSICAL QUANTITY

The quantities which can be measured by an instrument and by means of which we can describe the laws of physics are called physical quantities.

 

 

Fundamental quantities:

Although the number of physical quantities that we measure is very large, we need only a limited number of units for expressing all the physical quantities since they are interrelated with one another. So, certain physical quantities have been chosen arbitrarily and their units are used for expressing all the physical quantities, such quantities are known as

Fundamental, Absolute or Base Quantities (such as length, time and mass in mechanics)

(i)  All other quantities may be expressed in terms of fundamental quantities.

(ii)  They are independent of each other and cannot be obtained from one another.

 

An international body named General Conference on Weights and Measures chose seven physical quantities as fundamental:

(1)  Length

(2)  Mass

(3)  Time

(4)  Electric current,

(5)  Thermodynamic temperature

(6)  Amount of substance

(7)  Luminous intensity.

  

Note:

These are also called as absolute or base quantities. In mechanics, we treat length, mass and time as the three basic or fundamental quantities.

Derived: Physical quantities which can be expressed as combination of base quantities are called as derived quantities.

 

MANITUDE:

Magnitude of physical quantity = (numerical value) × (unit)

Magnitude of a physical quantity is always constant.

It is independent of the type of unit.

 begin mathsize 12px style table attributes columnalign left end attributes row cell rightwards double arrow space numerical space value space proportional to 1 over unit end cell row cell or    straight n subscript 1 straight u subscript 1 =  straight n subscript 2 straight u subscript 2  =  constant end cell end table end style

UNIT:

Measurement of any physical quantity is expressed in terms of an internationally accepted certain basic reference standard called unit. The units for the fundamental or base quantities are called fundamental or base unit. Other physical quantities are expressed as combination of these base units and hence, called derived units.
A complete set of units, both fundamental and derived is called a system of unit.

 

3.1 Principle systems of unit

There are various system is use over the world:

CGS, FPS, SI (MKS) etc

Table 1: unit of some physical quantities in different system

 

Physical

Quantity

System

CGS

(Gaussian)

MKS

(SI)

FPS

(British)

Fundamental

Length

centimetre

meter

foot

Mass

gram

Kilogram

Pound

Time

second

second

Second

Derived

Force

Dyne

Newton->N

poundal

Work or

Energy

erg

joule->J

ft-poundal

Power

erg/s

Watt->W

Ft-ponudal/s

 

Supplementary units:

(1)  Plane angle: radian (rad)

(2)  Solid angle: steradian (sr)

* The SI system is at present widely used throughout the world. In IIT JEE only SI system is followed.

 

Definition of some important SI

Units

  1. Meter: 1 m= 1,650, 763.73 wavelengths in vaccum, of radiation corresponding to organ-red light of krypton-86.
  2. Second: 1 s=9,192, 631,770 time periods of a particular from Ceasium-133 atom.
  3. Kilogram: 1kg=mass of 1 litre volume of water at 4C
  4. Ampere: It is the current which when flows through two infinitely long straight conductors of negligible cross-section placed at a distance of one meter in vaccum produces a force of 2×10-7 N/m between them.
  5. Kelvin: 1 K = 1/273.16 part of the thermodynamic temperature of triple point of water.
  6. Mole: It is the amount of substance of a system which contains as many elementary particles (atoms, molecules, ions etc.) as there are atoms in 12g of carbon -12.
  7. Candela: It is luminous intensity in a perpendicular direction of a surface of  begin mathsize 12px style left parenthesis 1 over 600000 right parenthesis straight m squared end style of a black body at the temperature of freezing point under a pressure of 1.013× 105 N/m2.
  8. Radian: It is the plane angle between two radiia of a circle which cut-off on the circumference, an arc equal in length to the radius.
  9. Steradian: The steradian is the solid angle which having its vertex at the centre of the sphere, cut-off an area of the surface of sphere of sphere equal to that of a square with sides of length equal to the radius of the sphere.

Significant digits, rounding off & mathematical operation, Types of Errors

Whenever an experiment is performed, two kinds of errors can appear in the measured quantity.

(1) Random and (2) systematic errors

  1. Random errors appear randomly because of operator, fluctuations in external conditions and variability of measuring instruments. The effect of random error can be somewhat reduced by taking.
    The average of measured values. Random errors have no fixed sign or size.
  2. Systematic error occur due to error in the procedure, or miscalibration of the instrument etc. Such errors have same size and sign for all measurements. Such errors can be determined.
    A measurement with relatively small random error is said to have high precision. A measurement with small random error and small systematic error is said to have high accuracy. The experimental error [uncertainty] can be expressed in several standard ways.

    Error limits Q ± ∆Q is the measured quantity and ∆Q is the magnitude of its limit of error. This expresses the experimenter's judgement that the 'true' value of Q lies between Q - ∆Q and Q + ∆Q. This entire interval within which the measurement lies is called the range of error. Random errors are expressed in this form.

    Absolute Error

    Error may be expressed as absolute measures, giving the size of the error in a quantity in the same units as the quantity itself. Least Count Error: - If the instrument has known least count, the absolute error is taken to be half of the least count unless otherwise stated. Error may be expressed as relative measures, giving the ratio of the quantity’s error to the quantity itself. In general,

     begin mathsize 12px style relative space error equals fraction numerator absolute space error space in space straight a space measurement over denominator size space of space the space measurement space end fraction end style

    We should know the error in the measurement because these errors propagate through the calculations to produce errors in results.

    1. Systematic errors: They have a known sign. The systematic error is removed before beginning calculations bench error and zero error are examples of systematic error.
    2. Random error: They have unknown sign. Thus they are represented in the form A ± a. Here we are only concerned with limits of error. We must assume a “worst-case” combination. In the case of subtraction, A – B, the worst-case deviation of the answer occurs when the errors are either + a and – b or – a and + b. In either case, the maximum error will be (a + b).

    For example in the experiment on finding the focal length of a convex lens, the object distance (u) is found by subtracting the positions of the object needle and the lens. If the optical bench has a least count of 1 mrn, the error in each position will be 0.5mm. So, the error in the value of u will be 1 mm.

    Addition and subtraction rule:

    The absolute random errors add.

    Thus if R = A + B, r = a + b

    And

    If R = A – B, r = a + b

    Product and quotient rule: The relative random errors add.

     begin mathsize 12px style table attributes columnalign left end attributes row cell Thus space if space straight R equals AB ,   straight r over straight R equals straight a over straight A plus straight b over straight B end cell row cell and space if space straight R equals straight A over straight B comma then space also space straight r over straight R equals straight a over straight A plus straight b over straight B end cell end table end style

  3. Power rule: When a quantity Q is raised to a power P, the relative error in the result is P times the relative error in Q. This also holds for negative powers.
    begin mathsize 12px style text If R=Q end text to the power of straight p comma straight r over straight R equals straight P cross times straight q over straight Q end style
  4. The quotient rule is not applicable if the numerator and denominator are dependent on each other.
    E .g. if begin mathsize 12px style text R= end text fraction numerator XY over denominator straight X plus straight Y end fraction end style. We cannot apply quotient rule to find the error in It Instead we write the equation as follows begin mathsize 12px style 1 over straight R equals 1 over straight x plus 1 over straight Y end style  . Differentiating both the
    Sides, we get
    begin mathsize 12px style text-end text dR over straight R squared equals negative dX over straight X squared minus dY over straight Y squared.    Thus straight r over straight R squared equals straight x over straight X squared plus straight y over straight Y squared end style

SIGNIFICANT DIGITS

Significant figures are digits that are statistically significant. There are two kinds of values in science:

  1. Measured Values
  2. Computed Values

The way that we identify the proper number of significant figures in science are different for these two types.

MEASURED VALUES

Identifying a measured value with the correct number of significant digits requires that the instrument’s calibration be taken into consideration. The last significant digit in a measured value will be the first estimated position. For example, a metric ruler is calibrated with numbered calibrations equal to 1 cm. In addition, there will be ten unnumbered calibration marks between each numbered position. (Each equal to 0.1 cm). Then one could with a little practice estimate between each of those marking. (Each equal to 0.05 cm). That first estimated position would be the last significant digit reported in the measured value. Let’s say that we were measuring the length of a tube, and it extended past the fourteenth numbered calibration half way between the third and fourth unnumbered mark. The metric ruler was a meter stick with 100 numbered calibrations. The reported measured length would be 14.35 cm. Here the total number of significant digits will be 4.

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