Question

derive the expression for error analysis, when the quantity is expressed as the multiplication of two measured quantity

Answer 1

Answer:

This is easy: just multiply the error in X with the absolute value of the constant, and this will give you the error in R: If you compare this to the above rule for multiplication of two quantities, you see that this is just the special case of that rule for the uncertainty in c, dc = 0.

Answer 2

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derive the expression for error analysis, when the quantity is expressed as the multiplication of two measured quantity.

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Suppose a result X is obtained by the product of two quantities say a & b .

X = a + b ..............(i)

Let ∆a & ∆b are absolute error in the measurement of a and b and ∆X be the corresponding absolute error in X .

➡ X ± ∆X = ( a ± ∆b ) × ( b ± ∆b )

➡ X ± ∆X = ab ± a∆b ± b∆a ± ∆a∆b

➡ X ± ∆X = X ± a∆b ± b∆a ± ∆a∆b

➡ ∆X = ± a∆b ± b∆a ± ∆a∆b ...........(ii)

Divide equation (ii) by (i) we have,

± ∆X/X = ± b∆a/ab ± a∆b/ab ± ∆a∆b/b

The quantities ∆a/a , ∆b/b and ∆X/X are called relative errors in the values of a, b and X respectively. The product of relative error in a and b

∆a∆b is very small hence is neglected.

➡ ± ∆X/X = ± ∆a/a ± ∆b/b

➡ ∆X/X = ∆a/a ± ∆b/b

Hence maximum relative error in X = maximum relative error in 'a' + maximum relative error in'b'

The maximum℅ error in X = maximum℅ error in a + maximum℅ error in b .

Those when a result involves the product of two observed quantities, the relative error in the result is equal to the sum of the relative error in the observed quantities.

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