Question 1

For a circular channel flowing partially full, select the correct set of design facts: depth for maximum velocity, hydraulic mean depth for maximum velocity, and depth for maximum discharge.

Accepted Answer

Introduction / Context: Hydraulic proportioning of circular channels (sewers, tunnels) uses known optima for velocity and discharge. These conditions depend on the filling depth relative to diameter and on how area and wetted perimeter combine to produce hydraulic radius and velocity under a given slope and roughness. Given Data / Assumptions: Circular section of diameter D, flowing partially full under gravity. Chezy/Manning relations apply; roughness and slope are fixed during comparisons. Comparison across relative depths only. Concept / Approach: Velocity depends on hydraulic radius R and hydraulic radius increases with certain filling depths. Discharge depends on both area and velocity. Standard derivations using calculus on the geometric relations for A(θ) and P(θ) yield approximate optima: y/D ≈ 0.81 for maximum velocity, R ≈ 0.30 D at that state, and y/D ≈ 0.95 for maximum discharge. Step-by-Step Solution: Express A and P in terms of central angle and y/D.Compute R = A/P and use V ∝ R^(2/3) S^(1/2) (Manning) or V ∝ C √(R S).Differentiate with respect to y/D to locate maxima for V and Q = A V. Verification / Alternative check: Published design charts and handbooks confirm the canonical values y/D ≈ 0.81 for maximum V and y/D ≈ 0.95 for maximum Q for circular sections. Why Other Options Are Wrong: Other numerical sets do not match the known extrema and would lead to under- or over-sized channels. Common Pitfalls: Assuming the same depth maximizes both velocity and discharge—these are different optima. Using full-pipe formulas for partially full conditions. Final Answer: Depth for maximum velocity ≈ 0.81 D; hydraulic mean depth ≈ 0.30 D; depth for maximum discharge ≈ 0.95 D

Question 2

Head loss due to a sudden enlargement in a pipe (velocities v1 before and v2 after enlargement) is given by:

Accepted Answer

Introduction / Context: Energy losses at geometric discontinuities (like sudden enlargement) are major components of minor losses in pipe systems. Correctly recalling the Carnot–Borda formula helps estimate additional head needed and pump sizing. Given Data / Assumptions: Incompressible flow of a liquid (e.g., water). Steady flow with a sudden area increase from section 1 to section 2. Negligible elevation change between sections. Concept / Approach: Downstream of a sudden expansion, the separated eddies dissipate kinetic energy. The loss equals the difference between the incoming kinetic energy and the recoverable kinetic energy associated with the uniform downstream velocity distribution, which leads to a loss proportional to the square of the velocity difference. Step-by-Step Solution: Start with energy grade: h_L = (α1 v1^2 − α2 v2^2) / (2 g) − pressure recovery term.For sudden expansion with uniform profiles, the classical result simplifies to h_L = (v1 − v2)^2 / (2 g).This is the Carnot–Borda loss expression. Verification / Alternative check: Setting v1 = v2 gives zero loss as expected; a large expansion (v2 ≈ 0) yields h_L ≈ v1^2/(2 g), matching the intuition of losing nearly all incoming kinetic energy. Why Other Options Are Wrong: (v1^2 − v2^2)/(2 g) or (v2^2 − v1^2)/(2 g): these are simple kinetic energy differences, not the correct dissipation measure.(v1 + v2)^2/(2 g): unphysical; predicts excessive loss even if v1 ≈ v2. Common Pitfalls: Confusing sudden enlargement with sudden contraction loss formulas. Ignoring velocity coefficients (α) where profiles are highly non-uniform. Final Answer: (v1 - v2)^2 / (2 g)

Question 3

Classify the flow statements: incompressible flow, compressible flow, and uniform flow. Which combined statement is correct?

Accepted Answer

Introduction / Context: Basic flow classifications appear in nearly every fluid mechanics problem. Correctly identifying whether a situation is incompressible, compressible, uniform, or non-uniform dictates which equations and simplifications are valid (e.g., constant density continuity vs. variable-density continuity). Given Data / Assumptions: General physical definitions only; no particular geometry is fixed. Newtonian fluids in typical engineering conditions. Definitions apply regardless of laminar or turbulent state. Concept / Approach: Incompressible: density ρ is constant (good for liquids and low-Mach gases). Compressible: density varies with position (or time). Uniform flow: at any instant, velocity is constant along the streamline or direction of flow; it may still vary across the cross-section in a strict sense, but in 1-D modeling we consider the mean value constant along x. Step-by-Step Solution: Check (a): constant density is the hallmark of incompressible modeling.Check (b): compressible flow allows ρ = ρ(x), ρ(y), etc.Check (c): uniform flow implies dV/dx = 0 (spatially), for steady analyses. Verification / Alternative check: Textbook definitions consistently match these statements; no contradictions arise among them, so the collective option is correct. Why Other Options Are Wrong: Selecting only one of (a), (b), or (c) omits the other true statements. Common Pitfalls: Confusing uniform (spatially constant) with steady (time-invariant). Applying incompressible assumptions at high Mach numbers. Final Answer: All of the above

Question 4

A spherical load of 900 kg rolls transversely across the deck of a 10,000 kg ship through 9.8 m. If the ship’s metacentric height (GM) is 5 m, estimate the heel angle caused by the shift (assume tanθ = (w * d) / (W * GM)).

Accepted Answer

Introduction / Context: Ship stability problems frequently involve heeling moments produced by moving loads on deck. The small-angle relation using metacentric height (GM) provides a quick estimate of heel angle and is widely used for preliminary checks. Given Data / Assumptions: Ship weight W = 10,000 kg (displacement mass, neglecting added mass). Moving load w = 900 kg shifts laterally by distance d = 9.8 m. Metacentric height GM = 5 m. Small heel angle, so tanθ ≈ θ in radians is acceptable, but we will compute θ via the tangent relation. Concept / Approach: For a weight shift on a floating body, heeling moment M = w * d. The restoring moment using metacentric stability is W * GM * tanθ. Equating moments gives tanθ = (w * d) / (W * GM). This relation is standard for small angles and assumes the metacentre remains near fixed vertically for small tilts. Step-by-Step Solution: Compute numerator: w * d = 900 * 9.8 = 8820 kg·m.Compute denominator: W * GM = 10,000 * 5 = 50,000 kg·m.tanθ = 8820 / 50,000 = 0.1764 → θ ≈ arctan(0.1764) ≈ 10.1 degrees.Convert to degrees–minutes: approximately 10° 5′. Verification / Alternative check: If one included the load in total weight (W + w), tanθ = (w d)/((W + w) GM) = 8820/(10,900*5) ≈ 0.162 → θ ≈ 9.2°, close but slightly smaller. Many exam problems use the simpler W in the denominator; the given options align with ≈ 10° 5′. Why Other Options Are Wrong: 10° 10′, 10° 15′, 10° 20′: larger than the calculated angle for the stated data and formula. Common Pitfalls: Misplacing units (using N rather than kg for mass-based form) and mixing g factors. Using GM in centimetres while d in metres. Final Answer: 10° 5′

Question 5

For steady laminar flow through a circular pipe, the kinetic energy correction factor (α) equals:

Accepted Answer

Introduction / Context: The kinetic energy correction factor α accounts for non-uniform velocity profiles when converting average velocity to actual kinetic energy of the flow. It is important in energy equation applications where velocity profiles deviate from uniform, such as laminar pipe flow. Given Data / Assumptions: Circular pipe, fully developed laminar flow. Parabolic velocity distribution (Hagen–Poiseuille profile). Steady, incompressible flow. Concept / Approach: α is defined by KE per unit weight = α * (V̄^2 / (2 g)), where V̄ is mean velocity. For a parabolic profile, kinetic energy is greater than if the same discharge moved uniformly at V̄, hence α > 1. Integration over the circular cross-section yields α = 2 for laminar flow. Step-by-Step Solution: Let local velocity u(r) = umax * (1 − (r/R)^2).Compute mean velocity V̄ = (1/A) ∫ u dA = umax/2.Compute ∫ u^3 dA and relate to V̄ to obtain α = (∫ u^3 dA)/(A V̄^3) = 2. Verification / Alternative check: For turbulent pipe flow with flatter profiles, α approaches 1.0–1.1. The contrast reinforces that laminar profiles have a larger α due to stronger non-uniformity. Why Other Options Are Wrong: 0.5, 1.0, 1.5, 2.5: do not match the integrated result for laminar parabolic distribution. Common Pitfalls: Confusing kinetic energy correction factor α with momentum correction factor β (which equals 1.33 for laminar pipe flow). Assuming α ≈ 1 for all cases; that is only near-true for turbulent flows. Final Answer: 2.0

Question 6

In Francis’s rectangular weir formula with end contractions, the effective length is reduced by how much per end contraction when the head over the sill is H?

Accepted Answer

Introduction / Context: When measuring discharge over a rectangular sharp-crested weir with end contractions, the jet converges laterally, effectively reducing the length through which flow passes. Francis’s empirical formula corrects the crest length to account for this contraction. Given Data / Assumptions: Rectangular sharp-crested weir with end contractions. Head over crest (sill) is H. Standard, fully aerated nappe; no submergence. Concept / Approach: Francis proposed using an effective length L_e = L − 0.1 n H, where L is the physical crest length and n is the number of end contractions (typically n = 2 for a weir contracting on both sides). Thus each end contraction reduces the length by 0.1 H. Step-by-Step Solution: Identify number of end contractions: n = 0, 1, or 2 depending on abutment conditions.Compute L_e = L − 0.1 n H.Therefore, the reduction per end is 0.1 H. Verification / Alternative check: Standard hydrometry references reproduce the same reduction term in Francis’s formula and compare favorably with laboratory data for common head ranges. Why Other Options Are Wrong: 0.2 H, 0.3 H, 0.4 H, 0.5 H exaggerate the contraction effect and would underpredict discharge. Common Pitfalls: Forgetting to set n properly (e.g., suppressed weir has n = 0). Using the reduction with a drowned nappe; different corrections apply. Final Answer: 0.1 H

Question 7

A thin rectangular orifice of width b and depth d connects two reservoirs with a difference in liquid levels H. Neglecting losses other than coefficient of discharge Cd, the theoretical discharge expression is:

Accepted Answer

Introduction / Context: Flow through thin-plate orifices is a staple topic in hydraulics. When the head difference across the plate is H and the orifice has width b and depth d (rectangular opening), the discharge is estimated from the product of effective area and theoretical jet velocity, corrected by the discharge coefficient Cd. Given Data / Assumptions: Thin, sharp-edged rectangular orifice of width b and depth d. Head difference across the plate is H. Coefficient of discharge = Cd accounts for contraction and velocity effects. Concept / Approach: The theoretical velocity of efflux under head H is v_th = sqrt(2 * g * H). The geometrical area of the opening is A = b * d. Actual discharge equals Q = Cd * A * v_th = Cd * b * d * sqrt(2 * g * H). Step-by-Step Solution: Compute area: A = b * d.Compute theoretical jet speed: v_th = sqrt(2 * g * H).Multiply and apply Cd: Q = Cd * b * d * sqrt(2 * g * H). Verification / Alternative check: Dimensional analysis confirms units: [Q] = L^3/T from (L * L) * sqrt(L/T^2) = L^2 * L^0.5 / T = L^2.5/T (wait), with Cd making no unit change. But since sqrt(2 g H) has units L/T and A has L^2, Q has L^3/T as required. Why Other Options Are Wrong: Cd * b * sqrt(2 g H): missing the depth d (area incomplete).Cd * b * d * (2 g H): missing the square root; would massively overpredict Q.Cd * b * d^2 * sqrt(2 g H): area wrongly taken as b d^2. Common Pitfalls: Using head to the center of orifice when the head varies significantly across depth; for small d relative to H, taking uniform H is acceptable, otherwise integrate over depth. Confusing Cd with Cc or Cv individually. Final Answer: Cd * b * d * sqrt(2 * g * H)

Question 8

Why is a triangular (V-notch) weir preferred over a rectangular notch?  In open-channel flow measurement, select the most appropriate reason set explaining why a triangular V-notch is often chosen instead of a rectangular notch, especially for small…

Accepted Answer

Introduction / Context: Weirs are standard devices for measuring open-channel discharge. Among common shapes, the triangular (V-notch) weir is widely preferred for laboratory, irrigation, and drainage applications where flows are relatively small and accuracy at low heads is important. Given Data / Assumptions: Sharp-crested notch/weir under free, fully aerated flow. Head (h) measured above the notch crest at a standard upstream distance. Coefficient of discharge approximately constant over typical operating heads for a V-notch of fixed angle (often 90°). Concept / Approach: For a sharp-crested triangular notch, the ideal discharge varies with h^(5/2): Q = (8/15) * C_d * √(2g) * tan(θ/2) * h^(5/2). The stronger head exponent (5/2) compared with the rectangular notch's 3/2 makes the V-notch more sensitive at low heads, improving practical accuracy. Only a single head measurement is needed, and one plate can cover a wide flow range by virtue of the notch geometry. Step-by-Step Solution: Recognize the application: accurate measurement at small discharges.Note that V-notch discharge Q ∝ h^(5/2), giving higher sensitivity.Operationally, only the head h is read; geometry is fixed.The same notch accommodates a wide flow range because rising head effectively opens a larger portion of the V. Verification / Alternative check: Compare percentage change in Q for a small change in h for V-notch vs rectangular notch; the higher exponent produces larger relative change, hence finer resolution. Why Other Options Are Wrong: Each partial statement is true; selecting only one omits other genuine advantages. The comprehensive correct choice is “all the above”. Common Pitfalls: Ignoring aeration (submergence reduces accuracy); reading head too close to the nappe; using the rectangular notch for very low flows where measurement resolution suffers. Final Answer: all the above

Question 9

Steady flow in open channels: which parameter must remain unchanged with time at a section?  In a steady open-channel flow (not necessarily uniform), identify which quantity does not change with time at any given cross-section.

Accepted Answer

Introduction / Context: Steady vs. uniform flow are often confused. In open-channel hydraulics, “steady” refers to no time variation at a point, while “uniform” refers to no spatial variation along the reach. Understanding this distinction is essential for correct modeling and instrumentation. Given Data / Assumptions: Single-phase, single-channel flow. Flow is steady (∂/∂t = 0 at a section) but not necessarily uniform along the channel. No lateral inflow between adjacent sections for the discharge comparison. Concept / Approach: In steady flow, discharge Q at any fixed section remains constant with time. However, depth y and mean velocity V can vary from section to section (non-uniform) even though their local values are time-invariant. Therefore, the quantity guaranteed not to vary with time at a section is the discharge (rate of flow) Q. Step-by-Step Solution: Define steady flow: ∂Q/∂t = 0 at a fixed section.Recognize that y and V may differ spatially due to slope, roughness, or geometry changes.Hence, pick “rate of flow (discharge)”. Verification / Alternative check: Continuity for a control volume without lateral inflow/outflow ensures Q entering equals Q leaving and remains constant in time under steady conditions. Why Other Options Are Wrong: Depth and velocity can change along the channel (non-uniform flow). “All of these” would require uniform conditions, which are not stated. Common Pitfalls: Assuming steady implies uniform; misinterpreting short-term fluctuations as unsteadiness when time-averaged flow is steady. Final Answer: rate of flow (discharge)

Question 10

Continuity equation applicability: required conditions for using the simple form A₁V₁ = A₂V₂  For the commonly used one-streamtube continuity form A₁V₁ = A₂V₂, the flow should be steady, incompressible, effectively one-dimensional, and have a unifor…

Accepted Answer

Introduction / Context: The general continuity equation expresses mass conservation. The simplified duct formula A₁V₁ = A₂V₂ is widely used in pipes and channels but requires specific assumptions to be valid without density terms or profile corrections. Given Data / Assumptions: Single fluid, no phase change across the section pair. No lateral inflows between the two cross-sections. Objective: conditions for A₁V₁ = A₂V₂. Concept / Approach: The integral continuity equation reduces to ρA₁V₁ = ρA₂V₂ for steady, one-dimensional flow. If the fluid is incompressible (ρ constant), this becomes A₁V₁ = A₂V₂. A uniform velocity profile across each section is implied when using the area-mean velocity V without additional correction factors. Step-by-Step Solution: Start from mass conservation: ∮ ρ V·n dA = 0 (steady).Assume one-dimensional streamtube so that V is normal and uniform on each section.For incompressible flow, ρ cancels, yielding A₁V₁ = A₂V₂. Verification / Alternative check: For compressible flow, the correct form is ρ₁A₁V₁ = ρ₂A₂V₂. With non-uniform profiles, include a profile factor or use the integral form directly. Why Other Options Are Wrong: Each single statement is necessary but not sufficient alone. The complete and correct choice is “all the above”. Common Pitfalls: Using A V = constant in compressible gases with significant density changes; neglecting profile correction factors in high-Re turbulent ducts. Final Answer: all the above

Question 11

Head loss due to friction: which mechanisms contribute to the energy loss?  In internal and open-channel flows, the frictional loss of head includes the energy dissipation arising from which physical effects?

Accepted Answer

Introduction / Context: Head loss quantifies mechanical energy degraded to heat through internal friction and chaotic motion. Correctly identifying the sources of frictional loss is essential for sizing pipes, pumps, and channels. Given Data / Assumptions: Single-phase Newtonian fluids. Fully developed sections for representative behavior. Concept / Approach: In laminar flow, energy is dissipated by viscous shear alone. In turbulent flow, additional dissipation arises from eddying motion—turbulent shear—which ultimately converts kinetic energy into heat at the smallest scales through viscosity. Thus, both viscosity and turbulence are responsible for frictional head losses across practical flow regimes. Step-by-Step Solution: Recognize laminar regime: losses from viscous shear (molecular momentum diffusion).Recognize turbulent regime: eddy viscosity and turbulent mixing enhance momentum transfer, increasing losses.Therefore, select “both (a) and (b)”. Verification / Alternative check: Compare Darcy–Weisbach friction factor: f_laminar = 64/Re (purely viscous), while turbulent correlations (e.g., Colebrook–White) capture roughness and turbulence effects. Why Other Options Are Wrong: Neither viscosity alone nor turbulence alone explains losses across all regimes; “none” is clearly incorrect. Common Pitfalls: Assuming turbulence “replaces” viscosity; actually viscous dissipation remains the final sink for kinetic energy. Final Answer: both (a) and (b)

Question 12

Nozzle size for maximum power transmission through a pipe  A nozzle is fitted to the end of a straight pipe of length 320 m and diameter 0.10 m. If the Darcy friction factor f = 0.01 for the pipe, determine the nozzle diameter that maximizes the pow…

Accepted Answer

Introduction / Context: When a nozzle is attached to a long pipe, enlarging the nozzle increases jet power (more discharge) but also increases pipe velocity and friction loss, reducing available head at the nozzle. There exists an optimal nozzle size that maximizes the jet power. Given Data / Assumptions: Pipe length L = 320 m, pipe diameter D = 0.10 m. Darcy friction factor f = 0.01 (Darcy–Weisbach). Steady, incompressible flow; negligible minor losses; jet coefficient ≈ 1. Concept / Approach: For a pipe with end nozzle, the optimal area ratio a/A that maximizes jet power satisfies a classical condition obtained by differentiating power with respect to nozzle area, leading to the relation (8 f L / D) * (a/A)^2 = 1, hence d = D / ( (8 f L / D)^(1/4) ). Step-by-Step Solution: Compute parameter: 8 f L / D = 8 * 0.01 * 320 / 0.10 = 256.Fourth root: 256^(1/4) = 4.Optimal nozzle diameter: d = D / 4 = 0.10 / 4 = 0.025 m = 2.5 cm. Verification / Alternative check: The same result is consistent with the classical criterion that maximum power occurs when head lost due to pipe friction equals one-third of the static head at the pipe entrance for the optimal area ratio. Why Other Options Are Wrong: 2.4, 2.6, and 2.7 cm correspond to area ratios that depart from the optimal a/A, yielding less jet power. Common Pitfalls: Confusing Darcy and Fanning friction factors (off by a factor of 4) or including minor losses without stating them, which would shift the optimum slightly. Final Answer: 2.5 cm

Question 13

Identify the incorrect statement about radial flow  Which of the following statements is NOT correct for radial flow fields (e.g., flow toward or away from a centerline)?

Accepted Answer

Introduction / Context: Radial flow occurs in pumps, wells, and around bluff bodies. It differs from rectilinear flow because streamlines converge toward or diverge from a point/axis, altering geometry and kinematics. Given Data / Assumptions: Axisymmetric, inviscid-style reasoning for basic identification. Flow direction is predominantly radial. Concept / Approach: In radial flow, streamlines are not parallel; they either converge or diverge, causing pressure and velocity to vary with radius. Thus, any statement implying “streamlines parallel” is inconsistent with the very definition of radial motion. Step-by-Step Solution: Check (a): variation with radius r is expected → correct.Check (b): direction predominantly radial → correct.Check (c): both inward and outward patterns exist → correct.Check (d): “streamlines parallel” contradicts radial geometry → incorrect statement. Verification / Alternative check: Potential-flow solution for a source/sink has V_r ∝ 1/r and streamlines radiating from/into the origin; clearly not parallel. Why Other Options Are Wrong: Options (a)–(c) accurately describe radial flow behavior. Common Pitfalls: Confusing “one-dimensional” with “parallel streamlines”; even simplified 1-D radial models do not imply parallel lines. Final Answer: Flow is one-dimensional with streamlines parallel

Question 14

Representative velocity to use in hydraulic engineering calculations  When solving flow problems in pipes and open channels, which velocity is typically used for continuity, energy, and momentum equations?

Accepted Answer

Introduction / Context: Hydraulic equations (continuity A V, energy head V^2/2g, momentum ρ Q V) rely on a representative velocity. Because real velocity profiles vary across a section, a defined average is used to ensure mass and momentum balances are satisfied. Given Data / Assumptions: Fully developed or developing internal/open-channel flows. Use of control-volume formulations in design practice. Concept / Approach: The correct representative velocity is the cross-sectional average, defined as V_avg = Q / A = (1/A) ∫ V_local dA. This ensures that continuity, energy, and momentum equations remain consistent. Centerline or wall velocities do not conserve mass or momentum when substituted directly without profile factors. Step-by-Step Solution: Identify governing balances where V appears (A V, V^2/2g, ρ Q V).Define V_avg from Q = ∫ V dA = A V_avg.Use V_avg in hydraulic formulae; apply correction coefficients if high accuracy is required for energy or momentum (α, β). Verification / Alternative check: For laminar pipe flow, centerline velocity = 2 V_avg; using centerline value would double the implied discharge, violating continuity. Why Other Options Are Wrong: Centerline/wall values are point velocities; their mean is arbitrary and not physically founded for balances. Common Pitfalls: Ignoring kinetic energy/momentum correction factors in strongly non-uniform profiles. Final Answer: average velocity of flow over the cross-section

Question 15

Rule of thumb for hydraulic jump length  For practical design estimates, the length of a hydraulic jump on a horizontal floor is approximately how many times the jump height?

Accepted Answer

Introduction / Context: Hydraulic jumps dissipate supercritical flow energy and stabilize downstream conditions. Designers need quick estimates of jump length for apron sizing and protection against scour. Given Data / Assumptions: Horizontal, prismatic rectangular channel. Subcritical tailwater permitting a stable jump. Empirical rule for preliminary design (not a strict analytical derivation). Concept / Approach: Experiments show that the roller length L_j correlates with the sequent-depth ratio and Froude number upstream. A robust field rule is L_j ≈ 5 to 7 times the jump height (difference between the sequent depths). This range covers common Froude numbers encountered in spillways and sluices. Step-by-Step Solution: Define jump height Δy = y₂ − y₁.Use empirical estimate: L_j ≈ (5 to 7) * Δy for typical Fr₁.Refine later with site-specific testing or detailed correlations if needed. Verification / Alternative check: Compare with design charts correlating L_j/Δy to Fr₁; for Fr₁ ≈ 2–6, values commonly lie in the 5–7 range. Why Other Options Are Wrong: 2–3 and 3–5 under-estimate the roller for many practical cases, risking inadequate apron length. Common Pitfalls: Ignoring tailwater control; applying the rule uncritically to sloping or rapidly varying channels without adjustment. Final Answer: 5 to 7 times its height

Question 16

Metacentric height from transverse load shift on a ship  A load of magnitude w is moved transversely by a distance d across the deck of a ship of displacement W, causing it to heel by an angle θ (small). Determine the metacentric height (GM) from th…

Accepted Answer

Introduction / Context: Transverse stability of ships is characterized by the metacentric height GM. A convenient onboard test uses a known weight shift across the deck to induce a small heel angle; from the observed angle, GM can be computed. Given Data / Assumptions: Ship displacement (weight) = W. Movable load = w shifted transversely by distance d. Heel angle θ is small (tan θ ≈ θ in radians), ensuring metacentric theory validity. Concept / Approach: Equate the heeling moment produced by the load shift to the righting moment provided by GM. Heeling moment = w * d. Righting moment for small angles = W * GM * tan θ. Setting them equal yields GM = (w * d) / (W * tan θ). Step-by-Step Solution: Compute heeling moment: M_h = w * d.Righting moment: M_r = W * GM * tan θ.Equate M_h = M_r → W * GM * tan θ = w * d.Solve: GM = (w * d) / (W * tan θ). Verification / Alternative check: Dimensional check: GM has dimensions of length; (w/W) is dimensionless; d/ tan θ has length → consistent. Why Other Options Are Wrong: Options (a), (c), and (d) invert factors improperly, giving wrong dimensions or scaling. Common Pitfalls: Using degrees directly inside tan without conversion when computing numerically; applying the formula at large heel angles where metacentric theory is invalid. Final Answer: GM = (w * d) / (W * tan θ)

Question 17

Why falling water drops become spherical in shape  Select the primary physical reason that small falling water droplets assume a nearly spherical shape.

Accepted Answer

Introduction / Context: Naturally forming droplets (rain, sprays) are commonly spherical when small. Understanding the dominant force helps in atomization, nozzle design, and environmental modeling. Given Data / Assumptions: Small droplets where aerodynamic deformation and inertia are modest. No electric fields or surfactants altering interfacial properties. Concept / Approach: A liquid surface possesses surface tension, an energetic penalty per unit area. For a given volume, the sphere has the minimum surface area. Therefore, surface tension drives a free droplet to become spherical to minimize total surface energy. Cohesion is the molecular origin of surface tension, but the macroscopic effect selecting the sphere is specifically surface tension. Step-by-Step Solution: Identify governing forces for a free droplet: inertia, gravity, aerodynamic drag, and surface tension.For small radii, Weber number We = ρ V^2 L / σ is small → surface tension dominates.Surface energy E_s = σ * area is minimized by a sphere for fixed volume. Verification / Alternative check: As drops grow larger or fall faster, they deviate from sphericity (oblate shapes, breakup) when We increases, confirming the role of surface tension vs inertia. Why Other Options Are Wrong: Adhesion pertains to unlike materials (liquid/solid/air interfaces) and is not the primary cause; viscosity resists rate of deformation but does not set the equilibrium shape; “cohesion” is microscopic—surface tension is the correct macroscopic descriptor. Common Pitfalls: Assuming all raindrops are spherical regardless of size; large raindrops become distorted and may break. Final Answer: surface tension minimizing surface area

Question 18

Chezy’s formula in open-channel hydraulics In steady uniform open-channel flow, Chezy’s equation V = C * sqrt(R * S) is primarily used to determine which quantity?

Accepted Answer

Introduction / Context: Chezy’s formula is one of the earliest empirical relations in open-channel hydraulics. It relates mean velocity to channel resistance and slope, and remains foundational for understanding modern equations such as Manning’s formula. Given Data / Assumptions: Prismatic open channel with steady, uniform flow. Hydraulic radius R = A / P (area over wetted perimeter). Energy slope S ≈ bed slope for uniform flow. Concept / Approach: Chezy’s equation is written as V = C * sqrt(R * S), where V is mean velocity, C is Chezy’s resistance coefficient (depends on roughness and Reynolds number), R is hydraulic radius, and S is energy grade line slope. It is explicitly a velocity formula for open channels, not closed full-flow pipes (though an analogous form can be used with care). Step-by-Step Solution: Identify target variable: mean velocity V in open channels.Use V = C * sqrt(R * S) with known C, R, S.Then discharge Q = A * V if required. Verification / Alternative check: Compare with Manning’s formula V = (1/n) R^(2/3) S^(1/2). For a given channel, C can be related to n and R, showing consistency between the two approaches. Why Other Options Are Wrong: (a) and (b) pertain to pipe flow; Chezy’s relation is standard for open channels. (d) is incorrect since the formula clearly provides velocity, and (e) is unrelated. Common Pitfalls: Confusing hydraulic radius with hydraulic depth; assuming C is constant across regimes; applying pipe friction concepts directly to free-surface flow without adjustments. Final Answer: velocity of flow in open channels

Question 19

Standard rectangular weir discharge relation The empirical formula Q = m · L · √(2 g) · H^(3/2) for the discharge over a sharp-crested rectangular weir was suggested by which researcher?

Accepted Answer

Introduction / Context: Sharp-crested weirs are among the most common flow-measurement devices in open channels. The classical discharge relationship for rectangular weirs is strongly associated with Francis, whose experiments established the widely used head–discharge law. Given Data / Assumptions: Sharp-crested rectangular weir with crest fully aerated. Head H measured upstream sufficiently far from the crest. Coefficient m accounts for approach velocity and other corrections. Concept / Approach: The generalized form Q = m · L · √(2 g) · H^(3/2) is known as the Francis formula. It captures the 3/2 power dependence of discharge on head for a rectangular weir, with L as effective crest length and m as an empirical coefficient (close to unity with proper conditions). Step-by-Step Solution: Recognize device: sharp-crested rectangular weir.Recall standard discharge law with exponent 3/2.Attribute to Francis based on historic calibration work. Verification / Alternative check: Cipolletti is linked to trapezoidal weirs with 1 horizontal to 4 vertical side slopes; Bazin and Rehbock provided other corrections, but the core 3/2 law for rectangular weirs is attributed to Francis. Why Other Options Are Wrong: (a), (c), and (e) are historically associated with different devices or correction schemes. (d) is incorrect since the attribution is well established. Common Pitfalls: Confusing Cipolletti’s trapezoidal weir with the rectangular Francis weir; neglecting aeration requirements near the nappe. Final Answer: Francis

Question 20

Pitot tube orientation effect If a Pitot tube is inserted with its nose facing downstream (away from the oncoming flow), how does the liquid column in the tube respond compared with the static level?

Accepted Answer

Introduction / Context: A Pitot tube measures stagnation (total) pressure when its opening faces directly into the flow. Orientation errors change what pressure is communicated to the tube, affecting the indicated head and thus the inferred velocity. Given Data / Assumptions: Incompressible, steady flow. Pitot tube connected to a manometer referencing static level. Tip oriented downstream instead of upstream. Concept / Approach: With the tip facing upstream, the local dynamic pressure converts to a rise corresponding to stagnation pressure p_0 = p + ½ ρ v^2. If reversed to face downstream, the tube senses a pressure below static because the opening is in a separated wake with lower total pressure. The manometer column therefore falls relative to the static level. Step-by-Step Solution: Correct orientation: stagnation head h_0 = (p_0 − p_ref)/γ → column rises.Reversed orientation: sensed pressure drops below static → negative dynamic indication.Observed effect: column level falls compared with static reference. Verification / Alternative check: Wind-tunnel and pipe-flow practice confirm that a reversed Pitot registers sub-static pressure, sometimes approaching the suction in the wake, giving a depressed column. Why Other Options Are Wrong: (b) suggests no change, which is not what is observed; (c) is the normal upstream-facing behavior; (d) is unnecessary; (e) is not representative of steady readings. Common Pitfalls: Assuming Pitot works the same regardless of orientation; ignoring wake effects behind the probe. Final Answer: falls in the tube to a depth

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