The right limb of height h₂ in the simple u tube manometer containing mercury is open to atmosphere, while the left limb of height h₁ is connected to pipe in which a liquid of specific gravity 1 is flowing with uniform speed with h₁ < h₂. This condition gives ______ kind of pressure.
Try it first: pick an option, then read the explanation.
Asked in SSC JE 06/06/2024 (04pm-06pm)
Detailed Explanation
A simple U-tube manometer measures pressure in a pipe by balancing the fluid pressure against a column of heavy manometric liquid (such as mercury). The relative levels of mercury in the two limbs reveal whether the gauge pressure is positive or negative.
1. Establish the hydrostatic balance across the datum line \(X-X\) passing through the mercury meniscus in the left limb:
\[ p_{\text{pipe}} + \rho_1 g h_1 = p_{\text{atm}} + \rho_m g h_2 \]
2. Express in gauge pressure (\(p_{\text{gauge}} = p_{\text{pipe}} - p_{\text{atm}}\)):
\[ p_{\text{gauge}} = \rho_m g h_2 - \rho_1 g h_1 = g (\rho_m h_2 - \rho_1 h_1) \]
3. Given that \(\rho_m = 13,600\text{ kg/m}^3\), \(\rho_1 = 1,000\text{ kg/m}^3\), and \(h_2 > h_1\):
\[ \rho_m h_2 \gg \rho_1 h_1 \;\implies\; p_{\text{gauge}} > 0 \]
Hence, the pressure inside the pipe is strictly a positive gauge pressure.
Why “positive gauge pressure” Is Correct
Because the mercury level in the open right limb is pushed higher than the meniscus in the left limb (\(h_2 > h_1\)), the fluid pressure in the pipe easily overcomes atmospheric pressure and lifts the dense mercury column, confirming a positive gauge pressure.
Why the Other Options Are Wrong
- A. atmospheric pressure: Atmospheric pressure is the reference datum against which the manometer measures, not the pipe pressure itself.
- B. negative gauge pressure: Negative gauge pressure (vacuum) occurs when the mercury level in the left limb rises above the right open limb (h₁ > h₂).
- D. absolute pressure: Absolute pressure requires adding atmospheric pressure (p_abs = p_atm + p_gauge) to the manometer reading.
Key Point
In a U-tube manometer: Right limb higher (h₂ > h₁) = positive gauge pressure; left limb higher = vacuum.
Key Concepts for Students
- The horizontal datum X-X must be taken at the lowest interface between the pipe liquid and manometric fluid.
- Manometer equation: p_left = p_right along any continuous horizontal level within the same static fluid.
- Mercury (S = 13.6) is preferred because its high density keeps limb heights compact and manageable.
Practice more: this is question 1 in Measurement of Pressure MCQs – Part 1. Try the whole set and check every answer instantly.