Q1 50M Compulsory calculate Strength of materials and structural analysis
(a) An aluminium tensile specimen has a diameter of 30·50 mm and a gauge length 275 mm. If the force of 17·50 × 10⁴ N elongates the gauge length by 1·28 mm, determine the Poisson's ratio and the modulus of elasticity. Also, determine by how much the force causes the diameter of the specimen to contract. Assume shear modulus G = 22 GPa and yield strength σᵧ = 435 N/mm². (10 marks)
(b) A solid steel shaft of diameter 65 mm is to be designed using an allowable shear stress τₐₗₗₒw = 60 N/mm² and an allowable angle of twist per unit length θ = 1·05° per metre. Determine the maximum permissible torque that may be applied to the shaft. Take shear modulus as 80 GPa. (10 marks)
(c) A rigid box of mass 85 kg shown in the figure below rests on a floor. The coefficient of static friction for the contact surface is 0·25. What will be the maximum force, 'F' and the highest position, 'h' of its application so that the rigid box neither slides on the floor nor tips over? (10 marks)
(d) As shown in the figure, a beam of symmetrical I-section spanning 8·0 m is prestressed by a parabolic cable with an eccentricity of 150 mm at the centre of the span and zero at supports. The beam supports a uniformly distributed live load of 2·5 kN/m.
(i) Find the effective force in the cable for balancing the dead and live loads on the beam.
(ii) Calculate the shift of the pressure line from the tendon's centre line.
Take unit weight of concrete as 24 kN/m³.
(All dimensions are in mm) (10 marks)
(e) A tie member consisting of an ISA 75 × 50 × 8 (E 250 grade of steel) is connected to a 12 mm thick gusset plate using a 6 mm fillet weld at site. The welding is done on its three sides as shown in the figure. The angle between fusion faces is 75°. Find the lengths of weld L_w₁ and L_w₂, if the connection is designed to transmit a load equal to the design strength of the member. (10 marks)
For ISA 75 × 50 × 8, A_g = 938 mm² and C_xx = 25·2 mm
Take γ_mo = 1·10 and for site welding, γ_mw = 1·5.
K = 0·7 for 60° – 90° angle between fusion faces.
For E 250 grade steel :
f_u = 410 MPa
f_y = 250 MPa
Answer approach & key points
Calculate the required values for all five parts systematically, spending approximately 15% time on (a), 15% on (b), 20% on (c), 35% on (d) as it has two sub-parts with prestressing calculations, and 15% on (e). Begin each part with stating the relevant formula, show substitution with units, compute step-by-step, and conclude with the final answer and appropriate units. For parts (c) and (d) requiring figures, sketch clear free-body diagrams and cable profiles respectively.
- Part (a): Calculate longitudinal strain, lateral strain using G = E/[2(1+ν)] relationship, then find Poisson's ratio ν, modulus of elasticity E, and diameter contraction using correct sign convention
- Part (b): Determine maximum permissible torque by checking both shear stress criterion (τ = 16T/πd³) and angle of twist criterion (θ = T/GJ in rad/m), then select the governing lower value
- Part (c): Draw FBD showing weight, applied force F at height h, normal reactions, and friction; establish equilibrium equations for sliding condition (F = μW) and tipping condition (moment about leading edge), solve simultaneously for F_max and h_max
- Part (d)(i): Calculate dead load from I-section dimensions (to be assumed or standard ISHB), determine prestressing force P using load balancing concept where parabolic tendon provides upward uniform load w_up = 8Pe/L² balancing total downward load
- Part (d)(ii): Calculate shift of pressure line using the relationship between tendon eccentricity, prestress force, and applied moments, recognizing that pressure line shifts by M/P from tendon line
- Part (e): Calculate design strength of angle section T_d = A_g·f_y/γ_mo, determine design weld strength per mm, use equilibrium of moments about centroid to find L_w1 and L_w2 with throat thickness t_t = K·s = 0.7×6 mm
- For all parts: Maintain consistent units (N, mm, MPa or GPa), apply appropriate IS code provisions (IS 800:2007 for steel, IS 1343 for prestressing), and state assumptions clearly where dimensions are not explicitly given
Q2 50M solve Structural analysis and steel beam design
(a) Draw the shearing force and bending moment diagrams for the beam loaded as shown in the figure below. (8 marks)
(b) During the design of a beam, an ISMB 550 @ 1·037 kN/m is selected for use as a simply supported beam of 7 m span carrying a reinforced concrete floor capable of providing lateral restraint to the top compression flange. The total uniformly distributed load is made up of 100 kN dead load and 150 kN imposed load. In addition to this load, the beam also carries a point load at its midspan which is made up of 50 kN dead load and 50 kN imposed load. Check the adequacy of the section for the following :
(i) Shear strength
(ii) Bending strength
(iii) Deflection
(iv) Web buckling at support
Assume the section is plastic.
Given :
Stiff bearing length = 100 mm
f_y = 250 MPa, E = 2 × 10⁵ MPa
γ_mo = 1·1
For plastic section β_b = 1·0
For simply supported beam, ψ = 1·2
| KL/r | 90 | 100 | 110 | 120 |
|------|-----|------|------|------|
| f_cd (MPa) | 121 | 107 | 94·6 | 83·7 |
Properties of ISMB 550 :
Elastic section modulus, Zₑ = 2359·8 × 10³ mm³
Plastic section modulus, Zₚ = 2711·98 × 10³ mm³
Moment of Inertia about major axis, I₂₂ = 64900 × 10⁴ mm⁴
(All dimensions are in mm) (20 marks)
(c) Using the unit load method, determine horizontal and vertical components of deflection at point A for the frame loaded as shown in the figure below. Support C is fixed and B is a rigid joint. Take E as constant and same for both the members. (15 marks)
Answer approach & key points
Solve this multi-part structural analysis problem by allocating approximately 20% time to part (a) for SFD/BMD construction, 45% to part (b) for comprehensive steel beam design checks (shear, bending, deflection, web buckling), and 35% to part (c) for unit load method application. Begin with clear free-body diagrams, proceed through systematic calculations with IS 800:2007 provisions, and conclude with adequacy statements for each check.
- Part (a): Correct determination of support reactions and construction of SFD and BMD with proper sign conventions and peak values marked
- Part (b)(i): Shear strength check using V_d = V_n/γ_mo where V_n = 0.577f_yA_v with A_v = h×t_w for ISMB 550
- Part (b)(ii): Bending strength calculation using M_d = β_bZ_pf_y/γ_mo with lateral restraint condition (ψ = 1.2) and comparison with factored moment
- Part (b)(iii): Deflection check under service loads comparing calculated δ = 5WL³/384EI + PL³/48EI against span/360 or span/240 limits
- Part (b)(iv): Web buckling check at support using F_w = (b_1+n_1)t_wf_cd with n_1 = 2.5d and interpolation from given KL/r vs f_cd table
- Part (c): Application of unit load method with proper virtual work integration, calculation of horizontal and vertical deflection components at A using M and m diagrams
- Correct use of partial safety factors: 1.5 for dead load, 1.5 for imposed load, and load combinations per IS 800:2007
Q3 50M design RCC slab design, steel column design, structural analysis
(a) Design a floor slab to cover a room with internal dimensions of 4·5 m × 6·0 m. The slab is simply supported on all the sides on 230 mm thick masonry walls. The slab carries a live load of 4·0 kN/m² and a dead load due to finishing work of 1·0 kN/m². The corners of the slab are prevented from lifting up. Use M 20 concrete and Fe 415 steel. Assume mild exposure conditions. (20 marks)
Table : Bending Moment coefficients when four edges are discontinuous
| l_y/l_x | Short span coefficient, α_x | | | | | | Long span coefficient, α_y |
|---------|----------------------------|---|---|---|---|---|---------------------------|
| | 1·0 | 1·1 | 1·2 | 1·3 | 1·4 | 1·5 | for all values of l_y/l_x |
| α_x | 0·056 | 0·064 | 0·072 | 0·079 | 0·085 | 0·089 | 0·056 |
Modification Factor for Tension Reinforcement
Note : f_s is steel stress of service loads in N/mm²
f_s = 0.58 f_y (Area of cross-section of steel required)/(Area of cross-section of steel provided)
(b) A built-up column of effective length 10 m is designed by placing two ISMC 300 @ 363 N/m back to back at a spacing 'S' mm. The column is to carry a factored axial load of 1100 kN. Find the economical spacing 'S' of the two channel sections. Also design the batten system for the column. M 20 bolts of grade 4.6 are used for making the connections. Do not design the connections. Use E 250 grade of steel. (20 marks)
For connections : Edge distance = 32 mm
Gauge distance = 50 mm
Properties of ISMC 300 :
A = 4630 mm²
r_zz = 118 mm, r_yy = 26.0 mm
I_zz = 6420 × 10⁴ mm⁴
I_yy = 313 × 10⁴ mm⁴
C_y = 23.5
t_f = 13.6
300
z z
t_w=7.8
y
90
ISMC 300
(All dimensions are in mm)
(c) Using slope deflection method, determine the final end moments for the portal frame shown in the figure. The frame is fixed at A and D, and has rigid joints at B and C. Take EI as constant. (10 marks)
Answer approach & key points
Design requires systematic application of codal provisions across three distinct structural problems. Allocate approximately 40% time to part (a) slab design including load calculations, moment coefficients, and reinforcement detailing; 40% to part (b) built-up column covering spacing optimization and batten design; and 20% to part (c) slope deflection analysis with proper sign convention and equilibrium checks. Present each part sequentially with clear headings, showing all intermediate calculations before final design values.
- Part (a): Calculate effective spans, determine l_y/l_x ratio = 1.33, interpolate α_x = 0.0763, compute total load = 6.5 kN/m², design moments M_x and M_y, check depth for deflection using modification factor, calculate steel areas for both directions, and provide reinforcement detailing with bar diameter and spacing
- Part (b): Determine required spacing S by equating slenderness ratios about both axes (r_yy_modified ≈ r_zz), use Perry-Robertson formula or IS 800 buckling curves for E 250 steel, design batten system with spacing ≤ 1.5 times least r_yy of single channel, check batten strength for transverse shear and moment, and specify batten dimensions and connections
- Part (c): Identify degrees of freedom (θ_B and θ_C), write slope-deflection equations for members AB, BC, and CD considering fixed ends at A and D, apply joint equilibrium at B and C, solve simultaneous equations for unknown rotations, and compute final end moments with proper sign convention (clockwise positive)
- Correct application of IS 456:2000 for slab design including moment coefficients from Table 26 and deflection control through Clause 23.2.1
- Correct application of IS 800:2007 for built-up column design including Clause 7.6 for lacing and battening systems
- Proper use of given material properties: M 20 concrete (f_ck = 20 N/mm²), Fe 415 steel (f_y = 415 N/mm²), E 250 structural steel (f_y = 250 N/mm²)
Q4 50M solve Truss deflection, plastic analysis, water tank design
(a) Determine the horizontal component of deflection of joint D of the truss loaded as shown in the figure. The cross-sectional area of each member is tabulated below. Take E = 200 kN/mm². Length of the members are indicated in the figure. Use Castigliano's theorems. (15 marks)
Table : Area of cross-section
| S.No | Member | Area of cross-section |
|------|--------|----------------------|
| 1. | AB | 765 mm² |
| 2. | AD | 390 mm² |
| 3. | DB | 575 mm² |
| 4. | BC | 765 mm² |
| 5. | CD | 390 mm² |
(b) Determine the collapse load in case of propped cantilever of span 'l' and subjected to uniformly distributed load 'P' per metre length as shown in the figure. Take the plastic moment capacity of beam as M_P. (15 marks)
(c) A circular water tank with flexible base is to be designed for a capacity of 450 kL. The depth of water is to be 4 m including a free board of 250 mm. Find the dimensions of the tank and design and detail the wall of the tank. Use M 20 concrete and Fe 250 steel. (20 marks)
Given : Tensile stress in steel under direct tension for plain mild steel bars, σ_s = 115 MPa
Permissible direct tensile stress in concrete (M 20), σ_ct = 1·2 MPa
Unit weight of water, γ = 9800 N/m³
Answer approach & key points
Solve all three sub-parts systematically, allocating time proportional to marks: approximately 35% for part (a) truss deflection using Castigliano's theorem, 30% for part (b) plastic collapse analysis of propped cantilever, and 35% for part (c) complete water tank design including dimensioning, wall thickness calculation, and reinforcement detailing. Begin each part with clear identification of given data, show complete derivations with formulae, and conclude with boxed final answers. For part (c), present design calculations followed by a neat sketch showing reinforcement arrangement.
- Part (a): Correct application of Castigliano's first theorem by placing unit horizontal load at D, calculation of member forces P due to actual loading and ∂P/∂Q due to dummy load, summation of (P·L/AE)·(∂P/∂Q) for all members
- Part (b): Identification of correct collapse mechanism for propped cantilever with UDL—formation of plastic hinge at fixed end and within span, use of virtual work or equilibrium method to relate external work to internal energy dissipation
- Part (c): Calculation of tank diameter from capacity (450 kL) and effective depth (3.75 m), determination of hoop tension and bending moment coefficients from IS 3370 tables for H²/Dt ratio, design of vertical and horizontal reinforcement
- Part (c): Check for tensile stress in concrete against permissible σ_ct = 1.2 MPa, calculation of steel area using σ_s = 115 MPa, provision of minimum reinforcement as per IS 3370
- Part (c): Detailing showing distribution steel, vertical steel on both faces, development lengths, and proper lap splices for circular tank wall
Q5 50M Compulsory solve Fluid mechanics and geotechnical engineering problems
The flow rate of water over a weir is 3 m³/s. A 1 : 10 scale model of the weir is tested in a water channel. Answer the following :
(i) What flow rate should be used for the model ?
(ii) If a force of 15 N is experienced on the model, what force would be expected on the prototype ?
10
A rectangular wing on a small airplane has a 1·3 m chord and a 10 m span. When flying in air at 250 km/hour, the wing experiences a total aerodynamic force of 20 kN. If the lift to drag ratio is 3, what would be the lift coefficient of the wing ? Take density of air as 1·20 kg/m³.
10
An idealized radial turbine is rotating at 140 rev/min as shown in the figure. The absolute flow enters at 30° and leaves radially inward. The flow rate is 4·0 m³/s of water at 20°C. The blade thickness is constant at 10 cm. If density of water is 1000 kg/m³, what would be the theoretical power developed by the turbine ?
10
The shear stress induced at a depth of 7·0 m due to construction of a nearby foundation is 50 kN/m². The soil properties at the site are given below :
Unit weight (γ) = 18 kN/m³
Effective cohesion (C') = 12 kN/m²
Effective friction angle (φ') = 30°
Compute the factor of safety against shear failure assuming water table located far below the point. Also compute the percentage reduction in factor of safety if water table rises to the ground level.
Take unit weight of water = 9·81 kN/m³.
10
Excavation is made in a soil whose porosity is 35% and specific gravity of soil grains is 2·65. A 3·0 m layer of this soil is subjected to an upward seepage head of 4·0 m. What factor of safety exists against boiling (piping) ? If a factor of safety of 2 is required against boiling, what depth of gravel is required to be placed above the soil stratum ? Assume unit weight of gravel and the soil to be the same and loss of head in the layer to be negligible. Assume γw = 9·81 kN/m³.
Answer approach & key points
Solve all five numerical problems systematically, allocating approximately 20% time to each part. Begin with dimensional analysis for the weir model (parts i-ii), then proceed to aerodynamic lift coefficient, turbine power calculation, factor of safety in shear failure, and finally seepage/boiling analysis. Present each solution with clear problem identification, formula application, substitution, and final answer with units.
- Part (i-ii): Apply Froude's model law correctly — Qm/Qp = (Lr)^(5/2) = (1/10)^(5/2) = 0.00316, giving Qm = 9.49 L/s; Force ratio Fp/Fm = (Lr)^3 = 1000, giving Fp = 15,000 N
- Part (iii): Calculate lift coefficient using CL = 2L/(ρV²A) where L = 15 kN (from 3:1 ratio), V = 69.44 m/s, A = 13 m²; yields CL ≈ 0.40
- Part (iv): Apply Euler turbine equation P = ρQ(u1Vu1 - u2Vu2) with u2=0 (radial exit), u1 = ωr1, Vu1 = V1cos30°; requires geometric interpretation from figure for radius
- Part (v): Compute σ = γz + Δσ = 18×7 + 50 = 176 kPa; τf = c' + (σ-u)tanφ' = 12 + 176×tan30° = 113.6 kPa; FS = τf/τ = 113.6/50 = 2.27; for WT at GL, FS reduces to ~1.14 (50% reduction)
- Part (vi): Calculate critical hydraulic gradient ic = (Gs-1)/(1+e) = (2.65-1)/1.538 = 1.073; actual i = 4/3 = 1.333; FS = ic/i = 0.805 (unsafe); for FS=2, required gravel depth h gives total head loss 4m over (3+h)m with revised gradient
Q6 50M solve Open channel flow, fluid mechanics and retaining wall analysis
A flow of 9·0 m³/s occurs in a long rectangular channel of 3·0 m width with 1·5 m depth of water flow. There is a smooth constriction in the channel to 2·0 m width in the downstream direction. Answer the following :
(i) What depths are to be expected in and just upstream of the constriction, if losses are neglected ?
(ii) Classify the gradually varied flow profile upstream of the constriction, with proper justification.
15
A two-dimensional incompressible flow field is given by
V = 2xy î + (x² – y²) ĵ , where î and ĵ are the unit vectors along x and y
axes, respectively. Answer the following :
(i) Determine the magnitude and the angle the velocity vector makes
with x-axis at x = 3 m and y = 1 m.
(ii) Is the flow physically possible ? If so, determine an expression for
stream function.
(iii) What is the discharge between the streamlines passing through
(1, 0) and (0, 1) ?
(iv) Is the flow irrotational ? Justify your answer with appropriate
reasons.
15
A retaining wall is shown in the figure below :
Layer ①
γ = 17 kN/m³
φ' = 28°
C = 0
Ground Water Table
Layer ②
γsat = 20 kN/m³
φ' = 35°
C = 0
Assuming that the wall can yield sufficiently, determine the Rankine active force per unit length of the wall and also determine the location of the resultant line of action.
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 35% time to the open channel flow sub-parts (i)-(ii), 40% to the fluid mechanics sub-parts (iii)-(iv), and 25% to the retaining wall problem. Begin with clear identification of given data, apply relevant governing equations (specific energy, continuity, stream function theory, Rankine earth pressure), show all computational steps, and conclude with physical interpretation of results including flow classification and pressure distribution diagrams.
- For (i): Calculate critical depth (yc = 0.972 m) and specific energy (E = 2.028 m), then solve for conjugate depths using specific energy equation at constriction, identifying supercritical and subcritical alternatives
- For (ii): Classify the GVF profile as M1 curve with proper justification based on normal depth > critical depth and mild slope conditions upstream of constriction
- For (iii)-(iv): Verify continuity equation (∂u/∂x + ∂v/∂y = 0), derive stream function ψ = x²y - y³/3, calculate velocity magnitude (6.32 m/s) and angle (18.43°), compute discharge between streamlines (0.333 m³/s/m), and check irrotationality via vorticity (ωz = 0)
- For retaining wall: Calculate active earth pressure coefficients (Ka1 = 0.361, Ka2 = 0.271), determine effective stresses at layer interfaces and groundwater table, compute resultant force per unit length (≈ 180-200 kN/m), and locate centroid of pressure distribution
- Present clear free-body diagrams for the retaining wall showing pressure distribution with hydrostatic component below GWT and effective stress above
Q7 50M solve Soil mechanics and fluid mechanics problems
(a) A 3·0 m high sandy fill material was placed loosely at a relative density of 50%. Laboratory studies indicated that the maximum and minimum void ratios of the fill material are 0·90 and 0·52 respectively. Construction specifications required that the fill be compacted to a relative density of 80%. If Gs = 2·65, determine :
(i) Dry unit weight of the fill before and after compaction.
(ii) Final height of the fill after compaction.
Take γw = 9·81 kN/m³. (15 marks)
(b) A group of 9 driven cast in situ piles is installed in a layered cohesive soil deposit as shown in the figure below. Piles are 40 cm in diameter and 15 m long. The spacing between the piles is 1·2 m and the cutoff level is 2·0 m below the ground level. Determine the safe load of the piles with a factor of safety of 2·5. (15 marks)
(c) Glycerin is flowing through a 2·5 cm diameter horizontal pipe of 30 m length that discharges it into the atmosphere at 101 kPa. The flow rate through the pipe is 0·05 litres/second. Dynamic viscosity (μ) and density of glycerin are 0·25 kg/m-s and 1250 kg/m³, respectively.
Answer the following :
(i) What is the absolute pressure at 30 m length just before the exit of pipe ?
(ii) At what angle (θ) must the pipe be inclined downward from the horizontal for the pressure in the entire pipe to be atmospheric pressure and the flow rate to be maintained the same ? (20 marks)
Answer approach & key points
Solve all three numerical parts systematically, allocating approximately 30% time to part (a) on soil compaction, 30% to part (b) on pile group capacity, and 40% to part (c) on pipe flow hydraulics. Begin each part with stated assumptions and formulae, proceed through step-by-step calculations with proper units, and conclude with clearly boxed final answers for each sub-part.
- Part (a): Correct application of relative density formula to find void ratios before and after compaction, then dry unit weight using γd = Gs·γw/(1+e), and height reduction using mass conservation
- Part (a)(ii): Calculation of final height using relationship H2 = H1 × (1+e2)/(1+e1) based on constant mass and plan area
- Part (b): Determination of individual pile capacity in layered clay using α-method for skin friction and bearing capacity for base, then application of group efficiency factors for 3×3 pile group
- Part (b): Consideration of block failure mode versus individual pile failure for closely spaced piles (spacing/diameter = 3)
- Part (c)(i): Application of Hagen-Poiseuille equation for laminar flow to find pressure drop, verification of Reynolds number, and calculation of absolute pressure at pipe exit
- Part (c)(ii): Derivation of required inclination angle using energy equation where pressure gradient due to elevation head balances viscous losses, maintaining same flow rate
Q8 50M solve Soil mechanics and fluid mechanics problems
(a) At a site, fine sand exists to a depth of 10 m and below this lies a soft clay layer 7·0 m thick. Water table is 4·0 m below the ground surface. Saturated unit weight of sand is 20·0 kN/m³ and the wet unit weight above the water table is 18 kN/m³. The water content of the normally consolidated clay is 42%, liquid limit is 46% and the specific gravity of the solid particles is 2·75. The proposed construction will transmit a net stress of 130 kN/m² at the centre of the clay layer. Find the average settlement of the clay layer. (15 marks)
(b) A strip footing of width 2·8 m as shown in the figure is founded at a depth of 2·5 m below the ground surface in a C – φ soil. Water table is at a depth of 6 m below the ground surface. The average moist weight of soil above the water table is 18 kN/m³. Determine the ultimate bearing capacity, net ultimate bearing capacity, net allowable bearing pressure and the load/m for a factor of safety of 2·5. Use the general shear failure theory of Terzaghi.
Given : For φ = 30°, Nc = 37·2
Nq = 22·5
Nγ = 19·7
What will be the percent decrease in ultimate bearing capacity if during the flooding, water level rises 2 m above around surface ? (15 marks)
(c) Water at 20°C flows through a pipe of inlet diameter of 10 cm and passes further through a circular nozzle of diameter 2·5 cm, exits into the air as a jet, and strikes a vertical plate as shown in the figure. A force, F = 100 N is required to hold the plate stationary. Assuming steady, frictionless, one-dimensional flow and densities of water and mercury as 1000 kg/m³ and 13550 kg/m³ respectively, answer the following :
(i) Determine the velocities at sections ① and ②.
(ii) Determine the mass flow rate of water.
(iii) Determine the mercury manometer reading 'h'. (20 marks)
Answer approach & key points
Solve all three parts systematically, allocating approximately 30% time to part (a) settlement calculation, 30% to part (b) bearing capacity with Terzaghi's theory, and 40% to part (c) fluid mechanics with continuity, momentum and manometry. Begin each part with clear identification of given data, apply relevant formulas with proper unit conversions, and conclude with boxed final answers. For part (c), solve sub-parts (i)-(iii) sequentially as they are interdependent.
- Part (a): Calculate initial effective stress at mid-clay layer using submerged unit weights; determine compression index Cc from liquid limit using Cc = 0.009(LL-10); compute settlement using ΔH = (CcH₀/1+e₀)log₁₀[(σ'₀+Δσ)/σ'₀]
- Part (b): Apply Terzaghi's general shear failure equation for strip footing: qu = cNc + γDfNq + 0.5γBNγ; calculate net ultimate and allowable bearing pressures; determine percent decrease when water table rises to ground surface using submerged unit weight
- Part (c)(i): Apply momentum equation F = ρQ(V₂-0) = ρA₂V₂² to find V₂, then continuity equation A₁V₁ = A₂V₂ to find V₁
- Part (c)(ii): Calculate mass flow rate ṁ = ρA₁V₁ = ρA₂V₂ using velocity from (i)
- Part (c)(iii): Apply Bernoulli's equation between sections ① and ② including manometer reading h with mercury-water interface; solve for h using pressure balance ρw(V₁²-V₂²)/2 = (ρm-ρw)gh
- For all parts: Show proper unit conversions (kN/m³ to kg/m³ where needed), use g = 9.81 m/s², and state all assumptions clearly