Question

What do you mean by propagation of errors? Explain the propagation of errors in addition and multiplication.

Answer 1

Propagation of error is defined as effect on function by a variable's uncertainty (error) . It is technique to find effect on function , when there variables uncertainty . Example :- F = ma , for finding error in F
we take, ∆F/F = ∆m/m + ∆a/a , this process is not other than propagation error .


Propagation error in addition : when any function y is given in such a way that it is sum of two variable x and z then, error in y can be measured by
dy = dx + dz .
example :- if l₁ = 5 ± 0.1 and l₂ = 10 ± 0.2
Then, L = ? If L = l₁ + l₂
so, L = (5 + 10) ± (0.1 + 0.2) = 15 ± 0.3

Propagation error in multiplication :- if any function y = xz
Then, taking log both sides,
logy = logxy = logx + logy
logy = logx + logy
Now, differentiate both sides,
dy/y = dx/x + dz/z
Hence, for finding error in y = xz
we have to use formula , Dy/y = dx/x + dz/z
example :- m = (5 ± 0.1 )kg , a = (10 ± 0.2) m/s² then, find F = ?
∵F = ma
so, ∆F/F = ∆m/m + ∆a/a
First find F,
F = 5 × 10 = 50 N
Now, ∆F/50 = 0.1/5 + 0.2/10
∆F = 50( 0.02 + 0.02) = 50 × 0.04 = 2

Hence, F = (50 ± 2)

Answer 2

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