Scientific Notation

Using a proper positive or negative power of 10, every large or small number "N" can be expressed in terms of a number "M" and a power of 10 is as follows.

                                               N= Mx10ⁿ

This representation is called scientific notation. Here “M” is such a number whose first digit from left to right is a non-zero digit and followed by the decimal point and rest of the digits of the given number” N”.

For Example.

This distance of earth from sun is under..

                          S=150,000,000,000m

                         S=1.5 x 10¹¹m

The mass of hydrogen atom is

                                   m= 0,000,000,000,000,000,000,000,000,001,673kg  

It can represented in scientific notation as,

                                 m= 1.673x10^-27kg

Convention for Indicating SI Units.

• ·         Full name of the unit does not begin with a capital letter even if named after a scientist e.g newton.
• ·         Symbol of unit named after a scientist has initial letters such as ‘N’ for newton.
• ·         Prefix should be written before unit and immediately close to it.
• ·         A combination of base units is written each with one space apart e.g, Kgms^-2
• ·         Compound prefix are not allowed e.g. 1 μμF may be written as 1pF.
• ·         When a multiple of a base unit is raised to a power. The power applies to the whole multiple and not just the base unit. Thus

                          (1Km) ² = (1x10³ m) ²

                                       (1Km) ² = 1 x 10⁶ m ²

• ·         In particle work measurement should be recorded in most convenient unit e.g. reading of micro meter, screw gauge may be recorded in ‘nm’ and mass of a calorimeter in grams (g).
• Conversion of units:
  Although SI units have been adopted, but conventional system of units are still being used. Therefore it is sometimes needed to convert a unit from one system to another. Conversion factors between a few units are given below.
       1 inch = 2.54cm
       1mile = 1.609km
       1m    = 3.281 ft
   1 pound force =4.448 newton
   1 hour              =3600s
  Errors and Uncertainties:
  An error is defined as the difference between the measured values and the actual value of a quantity. The error arising due to the natural imperfections of the experimenter, limitation of apparatus and changes in environment during the measurement is often called uncertainty. The uncertainty is usually described as an error in a measurement.
  Classification of Errors:
  There are three types of error in the measurement of physical quantities.
  1.       Personal error
  2.       Systematic error
  3.       Random error
  Personal Error
  Personal error can arise due to the tendency of a person to make readings in favor of one particular reading. The error can arise due to an incorrect method of reading a scale.
  Systematic Error
  This error is due to the fault in the measuring instrument. The fault can be incorrect calibration of the scale such as the ruler or a watch. The error may also arise due to zero ‘0’ error between scales as in vernier caliper. This type of error can be checked by using more accurate set of instruments.
  Random Error
  This type of error can arise due to accidental changes in the experimental conditions such as temperature, line voltage and humidity etc. maintaining strict control conditions in laboratory repeating the measurement several times and taking an average can reduce the effect of random error.
  
  
  Precision and Accuracy
  Precision stands for the magnitude of the error in a measurement.
  Accuracy:
  Accuracy stands for the relative error, that is, error divided by the measured quantity.
  Explanation
  A measure is never absolutely precise and accurate. There is always an error or uncertainty in the measurement.  Suppose we measure the width of our book with meter-ruler and find it to be 15.4 cm. The maximum possible error Is half a millimeter =0.5mm because meter is calibrated in millimeter
                                                                         OR
  Let the measured distance from a point to another point is 195km. As we measure the distance with speedometer of the vehicle, which indicates a minimum distance of 1 km on its dial. The maximum possible error is now half a kilometer.
  
  Magnitude of error in the first measurement is (0.05cm). It is smaller than the magnitude of error (0.5km) is the second measurement. So the first measurement is more precise than the which are considered here.
  Sum or Difference
                          Suppose two quantities 'a' and 'b' are measured and we need to add or subtract them to obtain the quantity 'Q'.
  that is                                  Q=a+b
  Then,
  Total uncertainty in Q= uncertainty in 'a' + uncertainty in 'b'
  Example
    Q may be the distance 'x' between two separate position measurements X₁ and X₂ say
  
  
                                              X_{1 = (4.5 + 0.1) cm}
    _&                                       X_{2 = (1.46 + 0.1) cm}
  _{The distance 'x' is then recorded as,}
                                 _x= X_2- X₁
                                             _{=(10.1+ 0.2) cm}
  Product and Quotient
  The percentage uncertainty in the final answer is equal to the sum of the separate percentage uncertainties.
  Example
  We calculate the resistance 'R' of a resistor by applying ohm's law 
                               V= IR
                               R= V/I
  The measured voltage 'V' and the measured current 'I' are 
                      V= (5.2+0.1) V
                       I= (0.84+0.05)A
  Percentage uncertainty for 'V'
                          V= 0.1 V/5.2 X 100%
                          V= 2%
  Percentage uncertainty for 'I'
                  I= 0.05A/0.84A X 100%
                  I= 6%
  Total Uncertainty in 'R' as obtained from dividing 'V' by 'I' is
                        (2% + 6%)= 8%
  Hence result is 
                                R= 5.2V/0.84A
                                  =6.19 VA^{-1 + 8%}
                                  ^{=6.19 ohm + 8/100 X 6.2ohm}
                             ^{R = 6.2 + 0.5}Ω

  Power of A Quantity

  A measured quantity appear in the form of a power in an expression, then uncertainty is multiplied by power factor.

  Example

  The volume 'V' of a sphere of radius 'r' is given

  
  

  V=4/3 X πr³

  Uncertainty in case in  'V' is 3x%. Uncertainty in 'r' precise instrument for its measurement. If the radius of a small sphere is 

  r=2.25 + 0.01 cm

  Absolute uncertainty = + Least count 

                                  =+ 0.01 cm

  Percentage uncertainty = + 0.01 X 0.01cm/2.25cm X 100%

                                     =0.4%

  Total percentage uncertainty in 'V'

  V= 3 X 0.4%

     =1.2%

  Thus the volume

  V= 4/3πr³

  V= 4/3π(2.25cm)3 + 1.2% uncertainty 

  V= 47.687 cm3 + 1.2/100 X 47.689 cm³

  V= 47.7 + 0.6 cm³

  Uncertainty In Average Value

  The uncertainty in average value of the quantity is calculated as fellows

  ·         Find the mean of the measured values

  ·         Find deviation of each measured value from the mean value

  ·         The mean derivation is the uncertainty in the average value

  
  

  Example

  Six readings of micrometer screw gauge to measure the diameter of a wire in mm are

  1.20, 1.22, 1.23, 1.19, 1.22, 1.21

  Then,

  

   Without regard to the sign, the deviations in mm are 0.01, 0.01, 0.02, 0.01,0.00

  

  Thus the result is recorded as

        Mean diameter= 1.21 + 0.01mm

  
  

  Uncertainty In Periodic Timing Experiments

  The uncertainty in the time period of a vibrating body is expressed by dividing the least count of the timing device by the number of vibrations.

  Example

  Time period of 30 vibrations of a simple pendulum recorded by a stopwatch accurate upto 0ne-tenth of a second is -----second measurement. As we can calculate accuracy as follows;
  

  

  
  

  The relative error in second measurement is much less than the first measurement. Hence 2^nd measurement is more accurate, although less precise that the first measurement.

  Precision of a measurement is determined by the instrument being used and accuracy of a measurement depends on the frictional uncertainty in that measurement.

  Smaller the least count of the measuring device the more precise is the measurement. Lesser the relative error more accurate is the measurement. 

  
  

  Assessment of uncertainty in the final result

  The result of an experiment is usually calculated from a mathematical containing different quantities measured. Following rules apply for estimation uncertainty or error for the simple cases