MPSC 2023 Prelims Paper-I · Question 38 of 100

This question was cancelled by the Commission. It has no correct answer, no option is keyed, and this card does not identify one — nothing below states which choice the printed expression belongs to, because the Commission withdrew the question and there is nothing to nominate. One observation about the printed line is worth making, because it shows why a candidate in the hall would have found the item unworkable: as typeset, its third term is a bare fraction with no physical quantity attached to it, so what is on the page is not a well-formed physical statement and cannot be matched against anything. A candidate meeting a stem in that condition during the examination has no way of knowing that it has been or will be withdrawn, and the only sound response is the general one — recognise quickly that the item as printed cannot be made to work, leave it, and spend the time on questions that can be answered. What remains worth having is the physics the question was drawn from, so the rest of this card teaches that, taking the four named ideas in the order the paper prints them and treating each entirely on its own terms. The equation of continuity is a statement of the conservation of mass in a flowing fluid. For a fluid of constant density flowing steadily along a tube, whatever mass enters one section in a given time must leave every other section in the same time, so the product of the cross-sectional area and the speed of flow has the same value everywhere along the tube. The visible consequence is that a fluid speeds up where its channel narrows, which is why a river runs fast through a gorge and why a thumb placed over the mouth of a hose sends the jet further. Bernoulli's principle is a statement of the conservation of energy for the steady flow of an ideal fluid along a streamline: the energy carried by a unit of the fluid is made up of the energy it holds by virtue of its pressure, the energy it holds by virtue of its height, and the energy it holds by virtue of its motion, and the sum of the three does not change along the streamline. Since the total is fixed, an increase in one part must be paid for out of another, and the celebrated consequence is that the pressure within a moving fluid falls where the fluid moves faster — the reasoning behind the lift of an aerofoil, the working of an atomiser and the design of the Venturi meter. Pascal's law belongs to fluids at rest rather than to fluids in motion. It states that pressure applied to an enclosed fluid is transmitted undiminished to every part of that fluid and to the walls of its container, which is what allows a small force on a narrow piston to raise a heavy load on a wide one, and it is the working principle of the hydraulic press, the hydraulic lift and the hydraulic brake. Euler's equation is the general equation of motion for a fluid without viscosity, obtained by applying Newton's second law to a small element of the fluid and setting the forces acting on it against its acceleration; it is the broad framework from which simpler results for particular cases of steady flow are derived. Hold the four apart by the question each answers — what is conserved in a flowing fluid's mass, what is conserved in its energy, how pressure behaves in a fluid at rest, and how Newton's second law applies to a fluid element — and the topic is secure whatever form a future question takes.

Fluid mechanics divides into the study of fluids at rest and fluids in motion, and the two halves are governed by different results. In a fluid at rest, pressure at a point acts equally in all directions, increases with depth in proportion to the density of the fluid and the depth below the surface, and is transmitted undiminished throughout an enclosed fluid when it is applied from outside — the last of these being Pascal's law, which converts a small force on a small piston into a large force on a large one and so underlies every hydraulic machine. Beside it sits Archimedes' principle, that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces, which governs flotation and the measurement of relative density. In a fluid in motion two conservation laws do most of the work. Mass is conserved, which for an incompressible fluid in steady flow means that the volume passing any section of a tube each second equals that passing any other, so speed and cross-sectional area vary inversely. Energy is conserved, which along a streamline in an ideal fluid ties together the pressure, the height and the speed, so that a change in any one of them is compensated by the others. Real fluids depart from the ideal in two ways: they have viscosity, which dissipates energy against the walls and within the fluid itself, and their flow ceases to be smooth above a certain speed and becomes turbulent, with eddies that carry energy away.

General science in MPSC papers examines fluid mechanics through named results rather than through calculation: what a named law states, which everyday device or natural effect it explains, and which law accounts for a described observation. The devices are as examinable as the laws — the hydraulic lift and brake, the atomiser and spray gun, the Venturi meter, the aerofoil, the hydrometer and the syringe — and each is worth attaching to the principle behind it, since the commission asks in both directions. A second habit worth building applies to any question that prints an expression or a diagram rather than a sentence: read what is actually on the page before deciding what it must mean. Papers are typeset from manuscripts and then reproduced for the examination hall, and stacked fractions, superscripts and Greek symbols are exactly what suffers in that process. Most of the time the printing is sound and the reading takes a moment. Occasionally it is not, and a candidate who has decided in advance what a printed line ought to say can argue himself into an answer the page does not support. The safer discipline is to judge each item on what it actually prints, and to move on without regret when what it prints cannot be worked with.

• The equation of continuity expresses the conservation of mass in a flowing fluid: for an incompressible fluid in steady flow the product of the cross-sectional area and the speed of flow is the same at every section, so the fluid moves faster wherever its channel narrows.
• Bernoulli's principle expresses the conservation of energy for the steady flow of an ideal fluid along a streamline, combining the contributions of pressure, height and speed; its best known consequence is that pressure falls where a fluid moves faster, which accounts for the lift of an aerofoil and the working of an atomiser.
• Pascal's law states that pressure applied to an enclosed fluid at rest is transmitted undiminished to every part of the fluid and to the walls of its container, and it is the working principle of the hydraulic press, the hydraulic lift and the hydraulic brake.
• Euler's equation is the equation of motion of a fluid without viscosity, obtained by applying Newton's second law to a fluid element, and it provides the general framework from which simpler results for particular steady flows are obtained.
• Archimedes' principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces, and it governs flotation and the measurement of relative density.

• Confusing the statement about a fluid's mass with the statement about its energy; the first fixes the relation between cross-sectional area and speed of flow, while the second relates pressure, height and speed along a streamline
• Applying a result that holds for fluids at rest to a fluid in motion, or the reverse, when the two halves of the subject are governed by different principles
• Treating a printed expression as authoritative when it is visibly defective, and then reasoning towards whichever answer an assumed correction would support
• Spending time in the examination hall trying to repair a stem that cannot be worked with as printed, instead of leaving it and returning only if time allows

Fluid mechanics reaches MPSC papers as short recall of named principles and of the devices that embody them. The commonest forms are a description of an everyday effect with a request for the principle responsible, the name of a principle with a request for the device it explains, and a statement of a law with a request for the name attached to it. Numerical work, where it appears at all, is confined to simple applications such as the pressure at a given depth or the force multiplication of a hydraulic press. Because the topic contains several closely related named results, the option set is usually built from that one family, so the discrimination has to come from knowing precisely what each result says rather than from recognising which chapter the question belongs to. Occasionally, as with this item, an expression rather than a sentence is printed, and a printed expression should be read for exactly what it contains and no more.

No directly related past PYQ was found.

• practice — not a real PYQ

  The hydraulic lift used to raise a motor car in a service station works on which of the following principles ?

  • (a)Archimedes' principle
  • (b)Pascal's law
  • (c)The equation of continuity
  • (d)Newton's law of cooling

  Answer(b) Pascal's law — pressure applied to an enclosed liquid is transmitted undiminished throughout it, so a modest force applied to a piston of small area produces the same pressure at a piston of large area and therefore a much larger force there. Archimedes' principle concerns the buoyant force on an immersed body, the equation of continuity concerns the speed of a fluid in a tube of varying cross-section, and Newton's law of cooling belongs to heat transfer rather than to fluids.

• practice — not a real PYQ

  When water flowing steadily through a pipe reaches a section where the pipe is narrower, what happens to the speed of the water ?

  • (a)It increases, because the same volume must pass every section in a given time
  • (b)It decreases, because the narrower section offers greater resistance
  • (c)It remains unchanged, since water is incompressible
  • (d)It falls to zero until pressure has built up behind the narrow section

  Answer(a) It increases, because the same volume must pass every section in a given time — this is the conservation of mass applied to a flowing fluid, and it requires the product of cross-sectional area and speed to be the same everywhere along the pipe, so a smaller area must be matched by a greater speed. It is the reason a jet from a hose travels further when the opening is partly blocked, and the reason a river quickens where its valley narrows.