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Measurement of Pressure Question: What is the difference between the real (absolute) pressure and the gauge pressure? The gauge pressure, pg: Relative pressure to the atmospheric pressure ( p g > 0 or p g < 0 , while pabs > 0 always) ∴ pabs = p g + patm > 0 Pressure measuring techniques: Manometry Ú Necessary information: Pressure in an incompressible fluid - Depending only on height or depth (vertical) 1. Barometer (Mercury tube): Measuring patm (Atmospheric p) patm = γh + pvapor In case of Mercury, pvapor = 0.16 Pa << patm = 101300 Pa ∴ patm ≈ γh e.g. Mercury: h = 760 mm = 14.7 in. patm = 760 mmHg = 760 Torr cf. For water: h ≈ 34 ft = 10.36 m ! 2. Piezometer Tube (Vertical tube: open at the top) Question: What is the pressure at A? ∴ p A = p1= γ 1h1 + po where po : Pressure at the top Let po be zero (Gauge pressure) ∴ p A = p1= γ 1h1 Ú Advantage: Simple and accurate Disadvantage: p A > patm (Why?) Only for relatively small p A - Reasonable h Gauge fluid: Liquid rather than gas 3. U-tube Manometer (Two kind of fluids) y Finding the pressure PA (Start at A → (1) → (2) → (3) → top) Increase by γ 1h1 No change Then, Decrease by γ 2 h2 No change p A + γ 1h1 − γ 2 h2 = ptop Let ptop be zero (gauge pressure), p A + γ 1h1 − γ 2 h2 = 0 & ∴ p A = γ 2 h2 − γ 1h1 Ú Advantages 1. Gauge fluid: Deferent from the container fluid 2. Applicable to both liquid and gas e.g. In case of gas: At the step (1) → (2), γ 1h1 : negligible (Why?) ∴ p A = γ 2 h2 3. Applicable to wide range of p A due to Adjustable h (How?) e.g. For very large (small) p A , use a heavy (light) gauge fluid Large (Small) γ 2 of a gauge fluid & Reasonable h2 Ú Useful application - Pressure difference between two containers Q. What is the pressure difference between A and B as shown? y Finding the pressure difference Δp = p A − p B (Start at A → (1) → (2) → (3) → (4) → (5) → B) Increase No change Decrease No change Then, p A + γ 1h1 − γ 2 h2 − γ 3h3 = p B ∴ Δp = p A − p B = γ 2 h2 + γ 3h3 − γ 1h1 No change 3) Inclined-Tube Manometer (Measuring small pressure change) y Finding the pressure difference Δp = p A − p B (Start at A → (1) → (2) → B) 1 2 3 Step 1 [A → (1)] : Increase by γ 1h1 Step 2 [(1) → (2)] : Decrease by γ 2l2 sin θ (Height h2 = l2 sin θ Step 3 [(2) → B] : Decrease by γ 3h3 Then, p A + γ 1h1 − γ 2l2 sin θ − γ 3h3 = p B ∴ Δp = p A − p B = γ 2l2 sin θ + γ 3h3 − γ 1h1 If pipes A and B contain gas ( γ 1h1 & γ 3h3 : Negligible), p A − p B = γ 2l2 sin θ or ∴ l2 = p A − pB γ 2 sin θ