Unit & Dimension, Measurement

Limitations of Theory of Dimensions

Although dimensional analysis is very useful but it is not universal, it has some limitatios as given below.

  1. The dimensional formula for many physical quantities is the same. For example the dimensional formula for work, energy and torque is Unit and Dimension
  2. Numerical constant (k), having no dimensions such 3/4, e, 2π etc, can not be deduced by the method of dimensions.
  3. The method of dimensions can not be used to derive relations other than produces of power functions. For example :
    Unit and Dimension and Measurement - class eleven, IIT, NIT, and other Engineering Entrance Examinations25
  4. The method of dimensions can not be applied to derive formula if in mechanics a physical quantity depends on more than three physical quantities.Example :
    Unit and Dimension can not be derived by theory of dimensions but its dimensional correctness can be checked.
  5. Even if a physical quantity depends on 3 physical quantities, out of which two have some dimensions, the formula can not be derived by theory of dimensions.

Errors in Measurement

The errors in the measurement of a physical quantity is the difference between the true value and the measurement value of the physical quantity.

i.e Error = True value - Measured value.

Suppose, Unit and Dimension is the error in measurement of x then

Fractional error Unit and Dimension and percentage error Unit and Dimension

Example : Suppose a quantity x depends on three quantities A, B and C, as: Unit and Dimension

If ΔA, ΔB, ΔC and ΔX are the errors in A, B, C and x respectively, then

Unit and Dimension

This given fractional error in x.

The maximum fractional error is obtained by taking all signs as positive.

Unit and Dimension

∴ Maximum percentage error in x is :

Unit and Dimension

Significant Figures

The measurement of any physical quantity by any instrument is not absolutey correct. The degree of accuracy is shown by the significant figures. Significant figures do not change if we measure a physical quantity in different units.

Example: Unit and Dimension

Now, Unit and Dimension both have three significant figures.

Rules for Significant Figures

Example:

(i) All non-zero digits are significant figures.
Unit and Dimension

(ii) All zeros occuring between non-zero digits are significant figures:
Unit and Dimension

(iii) All zero to the right of the last non zero digit are not significant.
Unit and Dimension

(iv) All zero to the right of a derived point and to the left of a non zero digit are not significant.
Unit and Dimension

(v) All zeros to the right of a decimal point and to the right of a no-digit are significant.
Unit and Dimension

Rounding off the Measurements

The certain rules are applied in order to rounding off the measurements :

  1. If the digit to be dropped is less than 5, then the preceding digit remains unchanged. For Example : The number 9.64 in rounded off to 9.6.
  2. If the digit to dropped is greater than 5, then the preceding digit is raised by 1. For Example : The 9.77 is rounded off to 9.8.
  3. The number 9.65 is rounded off to 9.6, the number 9.650 is rounded off to 9.6.
  4. The number 9.75 is rounded off to 9.8.

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