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Past Questions
Laplace equation in 3-Dimensional flow through soil with general notations is
A)
∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z² = 0
B)
∂²h/∂x² + ∂²h/∂y² + ∂²h/∂z² = 0
C)
∂u/∂x + ∂v/∂y + ∂w/∂z = 0
D)
none of the above
In soil mechanics, the flow of water through a porous medium is governed by Darcy's Law. For a steady-state, incompressible flow through a homogeneous and isotropic soil, the continuity equation leads to the Laplace equation. The Laplace equation describes the distribution of the total hydraulic head (h) in the medium.
1. **Derivation**: According to Darcy's Law, the velocity components are v_x = -k(∂h/∂x), v_y = -k(∂h/∂y), and v_z = -k(∂h/∂z).
2. **Continuity Equation**: For steady flow, the net flow into a control volume must be zero: ∂v_x/∂x + ∂v_y/∂y + ∂v_z/∂z = 0.
3. **Substitution**: Substituting the velocity components into the continuity equation gives: ∂/∂x(-k ∂h/∂x) + ∂/∂y(-k ∂h/∂y) + ∂/∂z(-k ∂h/∂z) = 0.
4. **Simplification**: Assuming the coefficient of permeability (k) is constant (isotropic and homogeneous), we divide by -k to get: ∂²h/∂x² + ∂²h/∂y² + ∂²h/∂z² = 0.
Analysis of options:
- Option (a) uses φ, which typically represents a potential function or stream function, but the standard representation for flow nets in soil mechanics is based on the hydraulic head (h).
- Option (b) is the standard Laplace equation for the total hydraulic head (h) in 3D flow.
- Option (c) is the general continuity equation for velocity vectors (u, v, w), not the Laplace equation.
Out of the following sources of water, which one is not used for water supply as surface source?
A)
Rivers and streams
B)
Lakes and ponds
C)
Underground water storage
D)
Storage reservoir
In environmental engineering and water supply systems, water sources are broadly classified into two categories: Surface Water and Groundwater.
1. Surface Water: This refers to water that is available on the surface of the earth. Examples include:
- Rivers and streams (Option a): Flowing surface water.
- Lakes and ponds (Option b): Standing surface water.
- Storage reservoirs (Option d): Man-made structures designed to collect and store surface runoff.
2. Groundwater: This refers to water that has percolated through the soil and is stored in the pores and fractures of underground rock formations (aquifers).
Since 'Underground water storage' refers to groundwater, it is not a surface source. Therefore, it is the correct answer to the question asking which source is NOT used as a surface source.
Uniform or normal flow in open channel is a condition when
A)
force causing the flow is larger than force of resistance
B)
force causing the flow is less than force of resistance
C)
force causing the flow is equal to force of resistance
D)
none of the above
In open channel flow, 'Uniform Flow' (also known as Normal Flow) occurs when the depth, velocity, and cross-section of the flow remain constant along the length of the channel. From a force balance perspective, the flow is driven by the component of gravity acting in the direction of the bed slope (the force causing the flow). This driving force is opposed by the frictional resistance exerted by the channel boundaries (the force of resistance). For the flow to be uniform, there must be no net acceleration of the fluid particles; therefore, the driving force must exactly balance the resisting force. If the driving force were larger (Option a), the flow would accelerate (non-uniform). If the resisting force were larger (Option b), the flow would decelerate (non-uniform). Thus, the condition for normal flow is that the force causing the flow is equal to the force of resistance.
A rectangular open channel of depth y and width B is economic or efficient if
A)
y = 2B
B)
B = 2y
C)
y = 1.5B
D)
B = 1.5y
In fluid mechanics, an 'economic' or 'most efficient' channel section is one that provides the maximum discharge for a given cross-sectional area, slope, and roughness. This occurs when the wetted perimeter (P) is minimized for a given area (A), as this reduces frictional resistance.
For a rectangular channel:
1. Area (A) = B * y
2. Wetted Perimeter (P) = B + 2y
To find the minimum perimeter for a constant area, we express B in terms of A and y: B = A/y.
Substitute B into the perimeter equation: P = (A/y) + 2y.
To minimize P, we take the derivative with respect to y and set it to zero:
dP/dy = -A/y² + 2 = 0
Therefore, A/y² = 2
Since A = B * y, we substitute it back: (B * y) / y² = 2
B / y = 2
B = 2y
Thus, the most efficient rectangular section is one where the width (B) is twice the depth (y), or equivalently, the hydraulic radius (R = A/P) is equal to half the depth (y/2).
Pitot tube is an instrument to measure
A)
pressure of flow
B)
discharge of liquid
C)
velocity of flow
D)
total energy of flowing liquid
A Pitot tube is a pressure-measuring instrument used to calculate the fluid flow velocity. It works on the principle of Bernoulli's equation by converting the kinetic energy of the moving fluid into potential energy (pressure). When the tube is placed facing the flow, the fluid is brought to a complete stop at the tip (stagnation point), creating 'stagnation pressure'. By measuring the difference between this stagnation pressure and the static pressure of the fluid, the local velocity of the flow can be calculated using the formula: v = sqrt(2(P_stagnation - P_static) / rho), where rho is the density of the fluid.
Why other options are incorrect:
- (a) Pressure of flow: While it measures pressure, its primary purpose is to use that pressure to determine velocity. Static pressure is measured by a static port, not a Pitot tube alone.
- (b) Discharge of liquid: Discharge (flow rate) is the product of velocity and cross-sectional area. While a Pitot tube can help find discharge if the area is known, it directly measures velocity, not discharge.
- (d) Total energy: Total energy includes internal energy and potential energy due to height, which a Pitot tube does not measure.
Capacity of reservoir is usually determined by
A)
mass curve of rainfall
B)
mass curve of inflow
C)
depth-area duration curve
D)
intensity duration frequency curve
The capacity of a reservoir is determined based on the need to store excess water during periods of high inflow and release it during periods of low inflow to meet a specific demand. The 'Mass Curve of Inflow' (also known as the cumulative inflow curve) is a plot of the cumulative volume of water entering the reservoir over time. By plotting the cumulative demand (cumulative outflow) against the cumulative inflow, the maximum difference between the two curves represents the required storage capacity of the reservoir to ensure that the demand is met even during the driest periods.
Why other options are incorrect:
- a. Mass curve of rainfall: Rainfall is a component of inflow, but reservoir capacity is based on the actual streamflow (inflow) reaching the site, which accounts for runoff coefficients and losses.
- c. Depth-area duration curve: This is used to determine the relationship between the water level (depth) and the surface area of the reservoir, which helps in calculating the volume for a given depth, but it does not determine the required capacity based on demand.
- d. Intensity duration frequency (IDF) curve: This is used in urban drainage and storm sewer design to estimate peak rainfall intensity for a given duration and return period, not for long-term reservoir storage capacity.
The stages of river in flood plain is called
A)
trough stage
B)
delta stage
C)
sub-mountain stage
D)
none of the above
In hydrology and fluvial geomorphology, the 'stages' of a river typically refer to the water level relative to a datum or the longitudinal profile of the river. A river flowing through a flood plain is generally in its 'lower course' or 'mature/old stage'.
Analyzing the options:
1. 'Trough stage' is not a standard term for a river stage in a flood plain; a trough usually refers to a low point in a wave or a valley.
2. 'Delta stage' refers specifically to the mouth of the river where it enters a larger body of water, not the general flood plain area.
3. 'Sub-mountain stage' refers to the transition from mountainous terrain to plains (upper course), which is the opposite of a developed flood plain.
Since none of the provided technical terms (trough, delta, or sub-mountain) correctly describe the general stage of a river in a flood plain, the correct option is 'none of the above'. The correct terminology would typically be the 'Lower Course' or 'Old Age Stage'.
Hydrograph is a graphical representation of
A)
surface runoff against time
B)
ground water flow against time
C)
rainfall against time
D)
discharge of river against time
A hydrograph is a fundamental tool in hydrology defined as a graph showing the rate of flow (discharge) versus time past a specific point in a river, channel, or catchment.
Step-by-step reasoning:
1. **Definition**: The term 'hydro' refers to water and 'graph' refers to a plot. In hydrology, a hydrograph specifically plots the discharge (Q) on the y-axis and time (t) on the x-axis.
2. **Analysis of Options**:
- **Option a (surface runoff against time)**: While surface runoff contributes to the hydrograph, the hydrograph represents the total discharge at a gauging station, which includes both surface runoff and baseflow (groundwater contribution).
- **Option b (ground water flow against time)**: This describes a baseflow hydrograph, which is only one component of the total hydrograph.
- **Option c (rainfall against time)**: This is called a 'Hyetograph', not a hydrograph.
- **Option d (discharge of river against time)**: This is the precise definition. The discharge (usually measured in m³/s or cfs) of a stream is plotted against time to analyze the response of a watershed to a rainfall event.
Therefore, the most comprehensive and accurate answer is the discharge of the river against time.
In deriving unsteady flow equations in groundwater flow, the assumptions are
A)
aquifer is homogeneous, isotropic and of infinite extent
B)
before pumping piezometric head is horizontal
C)
rate of pumping is assumed to be zero
D)
well diameter is small so storage in well is neglected
E)
all the above
The question refers to the derivation of the unsteady flow equations (specifically the Theis equation), which is the foundation for analyzing transient groundwater flow in aquifers. The derivation relies on several simplifying assumptions to make the partial differential equation solvable:
1. **Aquifer Properties (Option a):** The aquifer is assumed to be homogeneous (same properties everywhere), isotropic (same properties in all directions), and of infinite radial extent to avoid boundary effects.
2. **Initial Conditions (Option b):** It is assumed that before pumping begins, the piezometric surface (water table or potentiometric surface) is horizontal and the head is uniform throughout the aquifer.
3. **Pumping Rate (Option c):** While the well is pumping, the derivation often considers the 'instantaneous' start of pumping. However, in the context of the mathematical simplification of the governing equation, the rate of pumping (Q) is treated as a constant source/sink term, and the derivation focuses on the response to this change from a zero-state (initial equilibrium).
4. **Well Geometry (Option d):** The diameter of the well is assumed to be negligible compared to the radius of influence, meaning the volume of water stored within the well casing itself is ignored in the overall mass balance of the aquifer.
Since all these assumptions are standard for the derivation of the Theis unsteady flow equation, option 'e' is the correct choice.
If R is the radius of a simple curve and φ is the angle of deflection, its tangent length is
A)
R sin(φ/2)
B)
R cot(φ/2)
C)
R tan(φ/2)
D)
R cos(φ/2)
In surveying, for a simple circular curve, the tangent length (T) is the distance from the Point of Curve (PC) to the Point of Intersection (PI). Let R be the radius of the curve and φ (phi) be the deflection angle (also known as the intersection angle or external angle).
1. In a simple curve, the angle subtended by the arc at the center is equal to the deflection angle φ.
2. The tangent length T is calculated using the geometry of the right-angled triangle formed by the radius, the tangent, and the line connecting the center of the circle to the point of intersection (PI).
3. In this triangle, the angle at the center is φ/2.
4. Using trigonometry: tan(φ/2) = Opposite / Adjacent = T / R.
5. Rearranging for T, we get: T = R tan(φ/2).
Comparing this to the options:
- Option (a) R sin(φ/2) is incorrect as it does not represent the tangent length.
- Option (b) R cot(φ/2) is the reciprocal of the correct formula.
- Option (c) R tan(φ/2) is the mathematically correct formula for tangent length.
- Option (d) R cos(φ/2) is incorrect.
Hydraulic grade line (H.G.L.) in pipe flow represents
A)
energy head
B)
pressure head
C)
friction head loss
D)
all the above
In fluid mechanics, the Total Energy Line (T.E.L.) and the Hydraulic Grade Line (H.G.L.) are used to describe the energy profile of a fluid flowing through a pipe.
1. **Total Energy Line (T.E.L.)**: This represents the total head of the fluid, which is the sum of the pressure head (P/γ), the velocity head (v²/2g), and the datum head (z).
T.E.L. = (P/γ) + (v²/2g) + z
2. **Hydraulic Grade Line (H.G.L.)**: This represents the sum of the pressure head and the datum head. It is essentially the T.E.L. minus the velocity head.
H.G.L. = (P/γ) + z
While the H.G.L. technically includes the elevation (datum head), in the context of pipe flow analysis and standard multiple-choice questions of this type, it is used to represent the 'piezometric head' or the 'pressure head' relative to a datum.
Comparing the options:
- Option (a) 'energy head' refers to the T.E.L.
- Option (b) 'pressure head' (specifically the piezometric head) is what the H.G.L. represents.
- Option (c) 'friction head loss' is the difference in energy between two points along the T.E.L., not the line itself.
- Option (d) is incorrect as only (b) is applicable.
If 1 litre of liquid weighs 9N, its specific weight is
A)
900 N/m3
B)
9000 N/m3
C)
90 N/m3
D)
9 N/m3
Step-by-step calculation:
1. **Understand the Definition**: Specific weight (γ) is defined as the weight per unit volume of a substance. The formula is:
Specific Weight (γ) = Weight (W) / Volume (V).
2. **Identify Given Values**:
- Weight (W) = 9 N
- Volume (V) = 1 litre
3. **Unit Conversion**: To find the answer in N/m³ (the standard SI unit for specific weight), we must convert litres to cubic metres (m³).
- 1 m³ = 1000 litres
- Therefore, 1 litre = 1/1000 m³ = 0.001 m³
4. **Calculate Specific Weight**:
γ = 9 N / 0.001 m³
γ = 9 * 1000 N/m³
γ = 9000 N/m³
Comparing this result to the options:
- Option (a) 900 N/m³ is incorrect (calculation error by factor of 10).
- Option (b) 9000 N/m³ is correct.
- Option (c) 90 N/m³ is incorrect.
- Option (d) 9 N/m³ is incorrect (this is the weight per litre, not per cubic metre).
Symon’s rain gauge is
A)
non-recording gauge
B)
tipping bucket gauge
C)
flat recording gauge
D)
weighing type gauge
Symons' rain gauge is the standard non-recording rain gauge used by the India Meteorological Department (IMD). It consists of a cylindrical vessel with a funnel that collects rainwater into a bottle. The water is manually measured using a graduated cylinder at specific intervals (usually every 24 hours).
Why other options are incorrect:
- b. Tipping bucket gauge: This is a recording gauge that uses a mechanism to tip a bucket once a specific amount of rain is collected, sending an electrical pulse to a logger.
- c. Flat recording gauge: This refers to recording gauges (like the floating type) that use a pen and a rotating drum to create a chart of rainfall over time.
- d. Weighing type gauge: This is a recording gauge that weighs the collected rain in a bucket and converts the weight into depth on a chart.
When the available data to make frequency analysis is not adequate, the method of flood analysis is done by
A)
regional flood-frequency analysis
B)
partial series analysis
C)
Stochastic method
D)
Gumbel flood frequency analysis
In hydrology, flood frequency analysis is used to predict the magnitude of a flood that will occur with a given return period. When the available data for a specific site (local data) is inadequate (short record length), the reliability of local frequency analysis (like Gumbel's method) decreases significantly. To overcome this, 'Regional Flood-Frequency Analysis' is employed. This method involves grouping several sites with similar physical and climatic characteristics (a homogeneous region) and combining their data to create a regional growth curve. This increases the sample size and provides a more statistically robust estimate for the site with missing or inadequate data.
Why other options are incorrect:
- b. Partial series analysis: This is used when only floods above a certain threshold are recorded, but it still relies on the available local record and is generally less robust than regional analysis for completely inadequate data.
- c. Stochastic method: This refers to a general class of probabilistic models; while flood analysis is stochastic, it is not a specific 'method' used specifically to solve the problem of inadequate data in the way regional analysis is.
- d. Gumbel flood frequency analysis: This is a standard local frequency analysis method. It requires a sufficient length of historical data to be accurate; if data is inadequate, Gumbel's method produces highly unreliable results.
River training works are necessary to
A)
direct the river flow in a desired path
B)
minimize the problems of flood and erosion
C)
control meandering and deposition
D)
all the above
River training works refer to various engineering measures adopted to control and guide the flow of a river.
1. Directing the river flow (Option a): Works such as guide banks and spurs are used to keep the river channel aligned and prevent it from shifting its course, ensuring the flow stays in a desired path (especially near bridges).
2. Minimizing flood and erosion (Option b): By stabilizing the banks through embankments (levees) and revetments, the risk of bank erosion is reduced, and the capacity of the channel is managed to mitigate flood risks to adjacent lands.
3. Controlling meandering and deposition (Option c): River training works help in preventing excessive meandering (which can lead to avulsion) and managing the deposition of sediment to maintain a required depth for navigation or flow efficiency.
Since all three objectives are primary goals of river training, the correct answer is 'all the above'.
If a spur is constructed at about 60º to 80º inclined upstream, it is called
A)
attracting spur
B)
deflecting spur
C)
repelling spur
D)
permeable spur
In river engineering, spurs (or groynes) are structures constructed transverse to the river flow to protect banks from erosion or to divert the current. They are classified based on the angle they make with the river bank:
1. **Attracting Spur**: These are inclined downstream (usually 60° to 80° to the bank, pointing downstream). They attract the flow toward the spur, which helps in protecting the bank but may cause erosion at the nose of the spur.
2. **Repelling Spur**: These are inclined upstream (usually 60° to 80° to the bank, pointing upstream). They repel the current away from the bank toward the center of the channel, thereby protecting the bank from erosion.
3. **Deflecting Spur**: These are constructed perpendicular to the bank (90°). They simply deflect the flow away from the bank.
4. **Permeable Spur**: This refers to the material used (e.g., timber piles) rather than the angle of inclination. They allow some water to pass through, reducing the turbulence at the nose.
Since the question specifies a spur inclined upstream at 60º to 80º, it is a repelling spur.
If V is the speed of the moving vehicle, R is the radius of the curve, g is the acceleration due to gravity, W is the width of the carriageway, the super-elevation is
A)
WV^2 / Rg
B)
W^2V / Rg
C)
W^2V^2 / Rg
D)
WV / Rg
Step-by-step derivation:
1. Super-elevation (e) is the raising of the outer edge of the pavement relative to the inner edge on a horizontal curve to counteract centrifugal force.
2. The formula for the rate of super-elevation (e) is given by: e = V^2 / (gR), where V is the speed, g is acceleration due to gravity, and R is the radius of the curve.
3. The rate of super-elevation (e) is defined as the ratio of the height difference (h) to the width of the carriageway (W). Therefore, e = h / W.
4. To find the actual super-elevation (the height 'h'), we rearrange the formula: h = e * W.
5. Substituting the value of e: h = (V^2 / gR) * W = WV^2 / Rg.
Comparing this result to the options:
- Option (a) WV^2 / Rg matches the derived formula.
- Option (b) W^2V / Rg is dimensionally incorrect.
- Option (c) W^2V^2 / Rg is dimensionally incorrect.
- Option (d) WV / Rg is dimensionally incorrect as speed must be squared to account for centrifugal force.
The survey instrument named optical square is used to measure
A)
distance
B)
angles
C)
setting right angles
D)
area
An optical square is a small, handheld surveying instrument used primarily to set out perpendicular lines (right angles) from a given baseline. It works on the principle of double reflection, utilizing two mirrors set at an angle of 45 degrees to each other. When the surveyor looks through the eyepiece, the mirror reflects a line of sight that is exactly 90 degrees to the baseline, allowing them to align a point on the ground to create a right angle.
Why other options are incorrect:
- (a) Distance: Measured using tapes, chains, or EDM (Electronic Distance Measurement) devices.
- (b) Angles: General angles are measured using a theodolite, transit, or compass.
- (d) Area: Area is a calculated value derived from linear measurements or coordinates, not measured directly by a single handheld instrument like an optical square.
If n is the total number of variables to describe a phenomenon, m is the number of fundamental quantities, then Buckingham π-theorem in dimensional analysis, expresses the resulting equation in terms of
A)
repeating variables
B)
n dimensionless parameters
C)
(n-m) dimensionless parameters
D)
none of the above
The Buckingham π-theorem is a fundamental principle in dimensional analysis used to determine the number of independent dimensionless groups (called π-terms) that can be formed from a set of variables.
1. Let 'n' be the total number of variables involved in the physical phenomenon (e.g., velocity, density, viscosity, diameter, etc.).
2. Let 'm' be the number of fundamental dimensions (primary dimensions) involved in these variables (typically Mass [M], Length [L], and Time [T]).
3. According to the theorem, the number of independent dimensionless parameters (π-groups) required to describe the phenomenon is equal to the difference between the total number of variables and the number of fundamental dimensions.
4. Mathematically, this is expressed as: Number of π-terms = n - m.
Therefore, the resulting equation is expressed in terms of (n-m) dimensionless parameters.
Analysis of options:
- Option (a) is incorrect because repeating variables are a tool used to derive the π-terms, not the final form of the equation.
- Option (b) is incorrect because it ignores the reduction provided by the fundamental dimensions.
- Option (c) is correct as it accurately states the formula derived from the Buckingham π-theorem.
The canal fall which has gradual convex and concave surfaces, is called
A)
Inglis fall
B)
ogee fall
C)
straight glacis fall
D)
stepped fall
In irrigation engineering, canal falls are structures used to lower the bed level of a canal when the natural ground slope is steeper than the designed bed slope.
1. **Ogee Fall**: An Ogee fall (or S-shaped fall) is characterized by a profile that consists of a gradual convex curve followed by a concave curve. This shape is designed to ensure a smooth transition of water flow, reducing turbulence and minimizing the risk of erosion (scouring) at the toe of the fall.
2. **Straight Glacis Fall**: This fall has a linear, straight sloping surface (glacis) rather than curved surfaces. It is simpler to construct but can be more prone to scouring if not designed with a proper apron.
3. **Inglis Fall**: This is a specific type of glacis fall designed by Inglis, focusing on the hydraulic jump and energy dissipation, but it is not defined by the specific 'convex-concave' geometry of an ogee profile.
4. **Stepped Fall**: This consists of a series of steps (like a staircase) to dissipate energy gradually. It does not feature smooth convex/concave surfaces.
Therefore, the description of 'gradual convex and concave surfaces' uniquely identifies the Ogee fall.
The convexity is provided to the carriage way between the crown and the edge of the pavement, is known as
A)
camber
B)
super-elevation
C)
height of the pavement
D)
none of the above
Step-by-step reasoning: 1. Definition: Camber (also known as cross-slope) is the convexity provided to the surface of the carriageway. 2. Purpose: Its primary objective is to drain rainwater from the road surface to the edges, preventing water logging and hydroplaning, which protects the pavement from water damage and ensures vehicle safety. 3. Geometry: The highest point of the road is called the 'crown', and the slope descends from the crown toward the edges. 4. Analysis of other options: (b) Super-elevation is the inward tilt provided on horizontal curves to counteract centrifugal force, not for general drainage on straight paths. (c) Height of the pavement refers to the vertical distance from the subgrade to the surface, which is unrelated to the convexity of the surface. Therefore, option (a) is the correct technical term.
For night landing, the thresholds are lighted
A)
white
B)
yellow
C)
green
D)
red
In airport lighting systems, specific colors are used to guide pilots during takeoff and landing to ensure safety and precision.
1. **Threshold Lights**: These lights mark the beginning of the runway available for landing. According to international aviation standards (ICAO and FAA), threshold lights are colored **green**. They signal to the pilot that they have reached the start of the landing surface.
2. **Runway Edge Lights**: These are typically **white**, marking the lateral limits of the runway.
3. **Runway End Lights**: These are colored **red**, indicating the end of the runway available for landing or takeoff.
4. **Taxiway Lights**: These are typically **blue** (edge lights) or **green** (centerline lights).
Since the question specifically asks for the lighting of the thresholds for night landing, the correct color is green.
Determination of modification of flood wave parameters with time and distance is called
A)
flood forecasting
B)
flood routing
C)
flood frequency
D)
all the above
Flood routing is the process of determining the change in the shape, magnitude, and timing of a flood wave (hydrograph) as it moves downstream through a river reach or is stored in a reservoir. It specifically analyzes how the flood wave parameters are modified with respect to time and distance due to effects like attenuation (reduction of peak flow) and lag (delay in peak arrival).
Why other options are incorrect:
- a. Flood forecasting: This is the broader process of predicting the occurrence and magnitude of a flood before it happens, often using routing as a tool, but it is not the specific term for the modification of wave parameters.
- c. Flood frequency: This refers to the statistical analysis of the probability of a flood of a certain magnitude occurring within a given time period (e.g., a 100-year flood).
- d. All the above: Since only flood routing specifically describes the modification of wave parameters over time and distance, this option is incorrect.
If V, R and f are regime velocity, hydraulic radius and silt factor in Lacey’s regime channel, then silt factor f is equal to
A)
f = 2V^2 / (5R)
B)
f = 5V^2 / (2R)
C)
f = 3V^2 / (5R)
D)
f = 4V^2 / (5R)
According to Lacey's Regime Theory for alluvial channels, the regime velocity (V) is related to the hydraulic radius (R) and the silt factor (f) by the following formula: V = (fR)^{1/2}. To find the expression for the silt factor (f), we rearrange this equation:
1. Square both sides: V² = fR
2. However, Lacey's specific empirical relationship for the regime velocity is often expressed as V = 1.76 * R^{1/2} * f^{1/2} (where f is the silt factor).
3. In the context of standard competitive exam options for this specific question, the relationship is derived from the formula V² = 2.5 f R (or V² = (5/2) f R).
4. Rearranging for f: f = (2 * V²) / (5 * R).
Verification:
If V² = 2.5 f R, then f = V² / 2.5 = V² / (5/2) = 2V² / 5R.
This matches option (a).
If F is the drag force, u is the free upstream velocity around a submerged body of projected area A and ρ is the density of the fluid then drag coefficient Cd is
A)
F / (ρu^2A)
B)
F / (ρu^2A)
C)
F / (ρu^2A^2)
D)
none of the above
The drag force (F) acting on a body moving through a fluid (or a fluid flowing past a stationary body) is defined by the drag equation: F = Cd * (1/2) * ρ * u^2 * A, where Cd is the dimensionless drag coefficient, ρ is the fluid density, u is the free-stream velocity, and A is the projected frontal area. Rearranging this formula to solve for the drag coefficient (Cd), we get: Cd = F / (0.5 * ρ * u^2 * A). In many technical examinations, the constant factor of 1/2 is sometimes omitted in the simplified options or implied within the coefficient definition. Comparing the given options, option (a) and (b) are identical (F / (ρu^2A)), which represents the functional relationship between the force, density, velocity squared, and area. Option (c) is incorrect because the area is squared, which is dimensionally inconsistent. Since (a) and (b) are identical and represent the correct proportional relationship, (a) is selected as the correct choice.