Assertion (A) : In a well-type manometer, ratio of cross-sectional areas of well and the tube is not the factor that determines the scale of the manometer. Reason (R) : Accuracy of a manometer is not affected by the shape or size of the tubes.
(A) is false, but (R) is true.
Assertion — FALSE. In a well-type manometer the liquid displaced up the narrow tube must come out of the wide well, so the well level falls as the tube level rises. Equating the displaced volumes,
\(A_w h_w = A_t h_t \Rightarrow h_w = h_t\dfrac{A_t}{A_w}\)
and the true pressure head is the sum of both movements:
\(h = h_t\left(1+\dfrac{A_t}{A_w}\right)\)
So the area ratio does determine the scale — it is exactly the factor by which the scale must be compensated. Manufacturers either make \(A_w \gg A_t\) so that the correction becomes negligible, or shorten the scale graduations by the factor above so the reading can be taken from the tube alone. Either way, the ratio has been accounted for.
Reason — TRUE. Once the scale is correctly compensated, the accuracy of the instrument depends on the density of the manometric liquid and on gravity, through
\(P=\rho g h\)
not on the shape or bore of the tubes. This is the hydrostatic paradox: pressure at a depth depends only on the depth and the density, not on the shape of the containing vessel. (In a very narrow tube capillarity and meniscus errors do intrude, which is why manometer tubes are made wide enough to keep those effects small — but that is a practical limit, not a dependence of the principle on tube shape.)
So the assertion is wrong while the reason is right.
The distinction being tested is between the scale (which must allow for the falling well level, hence the area ratio) and the accuracy (which rests on liquid density and gravity alone).
Hence, (A) is false, but (R) is true.
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