Although dimensional analysis is very useful but it is not universal, it has some limitatios as given below.
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, is the error in measurement of x then
Fractional error and percentage error
Example : Suppose a quantity x depends on three quantities A, B and C, as:
If ΔA, ΔB, ΔC and ΔX are the errors in A, B, C and x respectively, then
This given fractional error in x.
The maximum fractional error is obtained by taking all signs as positive.
∴ Maximum percentage error in x is :
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:
Now, both have three significant figures.
Example:
(i) All non-zero digits are significant figures.
(ii) All zeros occuring between non-zero digits are significant figures:
(iii) All zero to the right of the last non zero digit are not significant.
(iv) All zero to the right of a derived point and to the left of a non zero digit are not significant.
(v) All zeros to the right of a decimal point and to the right of a no-digit are significant.
The certain rules are applied in order to rounding off the measurements :
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