त्रिकोणीय नोच के माध्यम से डिस्चार्ज में त्रुटि \(\left(\frac{\mathrm{dQ}}{\mathrm{Q}}\right)\)और हेड की माप में त्रुटि \(\left(\frac{\mathrm{dH}}{\mathrm{H}}\right)\) में _____ संबंध है।

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UKPSC JE Civil 8 May 2022 Official Paper-II
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  1. \(\frac{\mathrm{dQ}}{\mathrm{Q}}=\frac{3}{2}\left(\frac{\mathrm{dH}}{\mathrm{H}}\right)\)
  2. \(\frac{\mathrm{dQ}}{\mathrm{Q}}=\frac{5}{2}\left(\frac{\mathrm{dH}}{\mathrm{H}}\right) \)
  3. \(\frac{\mathrm{dQ}}{\mathrm{Q}}=\sqrt{\frac{5}{2}}\left(\frac{\mathrm{dH}}{\mathrm{H}}\right) \)
  4. \(\frac{\mathrm{dQ}}{\mathrm{Q}}=2\left(\frac{\mathrm{dH}}{\mathrm{H}}\right)\)

Answer (Detailed Solution Below)

Option 2 : \(\frac{\mathrm{dQ}}{\mathrm{Q}}=\frac{5}{2}\left(\frac{\mathrm{dH}}{\mathrm{H}}\right) \)
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Detailed Solution

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संकल्पना:

V नाॅच के लिए निस्सरण निम्न द्वारा दिया जाता है:

\(Q = \frac{8}{{15}}{C_d}\tan \frac{\theta }{2}\;\sqrt {2g} \;{H^{\frac{5}{2}}}\)

गणना:

दिया गया है:

V नाॅच में दाबोच्चता के मापन में त्रुटि 1% है अर्थात् \(\frac{{dH}}{H} = 1~\% \)

अब निस्सरण समीकरण का अवकलन 'H' के संबंध में करने पर हमें मिलता है:

\(Q = \frac{8}{{15}}{C_d}\tan \frac{\theta }{2}\;\sqrt {2g} \;{H^{\frac{5}{2}}}\)

\(Q = K{H^{\frac{5}{2}}}\)

जहाँ: \(K = \;\frac{8}{{15}}{C_d}\tan \frac{\theta }{2}\;\sqrt {2g} \)

\(dQ = K\;\frac{5}{2}\;{H^{\frac{3}{2}}}\;dH\)

\(K = \frac{Q}{{{H^{\frac{5}{2}}}}}\)

\(\frac{{dQ}}{Q} = \;\left( {\frac{{K\;\frac{5}{2}\;{H^{\frac{3}{2}}}\;dH}}{{K{H^{\frac{5}{2}}}}}} \right) = \frac{5}{2}\frac{{dH}}{H}\)

निस्सरण के मापन में त्रुटि​

i.e \(\frac{{dQ}}{Q} = 2.5\% \)

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