Page 40 - Class-11-Physics-Part-1_Neat

26 PHYSICS

(a) Error of a sum or a difference
How will you measure the length of a line?
Suppose two physical quantities A and B have
What a naïve question, at this stage, you might measured values A ± ∆A, B ± ∆B respectively
say! But what if it is not a straight line? Draw where ∆A and ∆B are their absolute errors. We
a zigzag line in your copy, or on the blackboard. wish to find the error ∆Z in the sum
Well, not too difficult again. You might take a Z = A + B.
thread, place it along the line, open up the We have by addition, Z ± ∆Z
thread, and measure its length. = (A ± ∆A) + (B ± ∆B).
Now imagine that you want to measure the
length of a national highway, a river, the railway The maximum possible error in Z
track between two stations, or the boundary ∆Z = ∆A + ∆B
between two states or two nations. If you take For the difference Z = A – B, we have
a string of length 1 metre or 100 metre, keep it Z ± ∆ Z = (A ± ∆A) – (B ± ∆B)
along the line, shift its position every time, the = (A – B) ± ∆A ± ∆B
arithmetic of man-hours of labour and expenses
on the project is not commensurate with the or, ± ∆Z = ± ∆A ± ∆B
outcome. Moreover, errors are bound to occur The maximum value of the error ∆Z is again
in this enormous task. There is an interesting ∆A + ∆B.
fact about this. France and Belgium share a
common international boundary, whose length Hence the rule : When two quantities are
mentioned in the official documents of the two added or subtracted, the absolute error in the
countries differs substantially! final result is the sum of the absolute errors
Go one step beyond and imagine the in the individual quantities.
coastline where land meets sea. Roads and rivers
have fairly mild bends as compared to a The temperatures of two
coastline. Even so, all documents, including our t Example 2.8
school books, contain information on the length bodies measured by a thermometer are 0
0
0
0
of the coastline of Gujarat or Andhra Pradesh, t = 20 C ± 0.5 C and t = 50 C ± 0.5 C.
1
2
or the common boundary between two states, Calculate the temperature difference and
etc. Railway tickets come with the distance the error theirin.
between stations printed on them. We have
‘milestones’ all along the roads indicating the Answer t′ = t –t = (50 C±0.5 C)– (20 C±0.5 C)
0
0
0
0
distances to various towns. So, how is it done? 2 1
0
0
One has to decide how much error one can t′ = 30 C ± 1 C t
tolerate and optimise cost-effectiveness. If you (b) Error of a product or a quotient
want smaller errors, it will involve high
technology and high costs. Suffice it to say that Suppose Z = AB and the measured values of A
it requires fairly advanced level of physics, and B are A ± ∆A and B ± ∆B. Then
mathematics, engineering and technology. It Z ± ∆Z = (A ± ∆A) (B ± ∆B)
belongs to the areas of fractals, which has lately
become popular in theoretical physics. Even = AB ± B ∆A ± A ∆B ± ∆A ∆B.
then one doesn’t know how much to rely on Dividing LHS by Z and RHS by AB we have,
the figure that props up, as is clear from the 1±(∆Z/Z) = 1 ± (∆A/A) ± (∆B/B) ± (∆A/A)(∆B/B).
story of France and Belgium. Incidentally, this
story of the France-Belgium discrepancy Since ∆A and ∆B are small, we shall ignore their
appears on the first page of an advanced Physics product.
book on the subject of fractals and chaos!
Hence the maximum relative error
mass density is obtained by deviding mass by ∆Z/ Z = (∆A/A) + (∆B/B).
the volume of the substance. If we have errors You can easily verify that this is true for division
in the measurement of mass and of the sizes or also.
dimensions, we must know what the error will
be in the density of the substance. To make such Hence the rule : When two quantities are
estimates, we should learn how errors combine multiplied or divided, the relative error in the
in various mathematical operations. For this, result is the sum of the relative errors in the
we use the following procedure. multipliers.

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