Sensitivity of rectangular-notch discharge to head-measurement error For a rectangular weir/notch where Q ∝ H^(3/2), the ratio of percentage error in discharge to percentage error in head measurement is

Introduction / Context:
Discharge over a rectangular notch is commonly expressed as Q = C_d b √(2 g) H^(3/2). Because Q depends on H to the 3/2 power, small errors in head gauging can translate into larger fractional errors in discharge. Quantifying this sensitivity is essential in flow measurement practice.

Given Data / Assumptions:

• Rectangular notch, free nappe conditions.
• Discharge varies as H^(3/2), other factors held constant.
• Small measurement errors (use differentials/relative errors).

Concept / Approach:

If Q ∝ H^n, then dQ/Q = n (dH/H). Here n = 3/2. Thus the ratio of fractional (percentage) errors is 3/2.

Step-by-Step Solution:

Verification / Alternative check:

A 1% head error yields approximately a 1.5% discharge error, matching field experience that accurate head measurement is critical.

Why Other Options Are Wrong:

(a), (b), and (d) understate the sensitivity; (e) overstates it for a rectangular notch.

Common Pitfalls:

Confusing rectangular notch (exponent 3/2) with triangular notch where exponent differs (5/2), changing sensitivity.

Final Answer:

3/2