Fluid flash cards

1. 1.Define pressure at a point in a fluid. Is it a scalar or vector?

   Pressure $P=\dfrac{F_{\perp}}{A}$ is the normal force per unit area. It is a scalar — at a point it acts equally in all directions. SI unit: pascal ($1\,\text{Pa}=1\,\text{N/m}^2$).

   Hint: Force per unit normal area.

2. 2.How does pressure vary with depth $h$ in a static fluid of density $\rho$?

   $P = P_0 + \rho g h$, where $P_0$ is the pressure at the surface. Pressure increases linearly with depth and is independent of the container's shape or cross-section.

   Hint: Weight of the fluid column above.

3. 3.What is gauge pressure versus absolute pressure?

   Gauge pressure is the pressure relative to atmosphere: $P_{gauge}=P_{abs}-P_{atm}$. Absolute pressure $P_{abs}=P_{atm}+\rho g h$. Gauge can be negative (partial vacuum).

   Hint: Gauge is measured above atmospheric.

4. 4.State the hydrostatic paradox.

   The pressure at the base of a liquid column depends only on the vertical depth $h$, not on the shape or total weight of liquid. Vessels of very different shapes but the same base area and liquid height exert the same force on the base.

   Hint: $P=\rho g h$ — depends on depth, not shape.

5. 5.Why are all points at the same horizontal level in a connected static fluid at the same pressure?

   In a static, connected fluid of uniform density, pressure depends only on vertical depth. If two points at equal height had different pressures, the net horizontal force would drive flow — contradicting the static condition.

   Hint: Equal depth ⇒ equal $\rho g h$.

6. 6.A U-tube holds two immiscible liquids of densities $\rho_1$ and $\rho_2$. What is the balance condition?

   Equating pressure at the common interface level: $\rho_1 g h_1 = \rho_2 g h_2$, so $\rho_1 h_1 = \rho_2 h_2$. The denser liquid stands lower.

   Hint: Match pressures at the bottom common level.

7. 7.State Pascal's law.

   A pressure change applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of the container.

   Hint: Pressure transmits equally throughout.

8. 8.How does a hydraulic lift multiply force? Give the relation.

   Same pressure on both pistons: $\dfrac{F_1}{A_1}=\dfrac{F_2}{A_2}$, so $F_2 = F_1\dfrac{A_2}{A_1}$. A small force on a small piston lifts a large load on a large piston.

   Hint: Equal pressure, unequal areas.

9. 9.In an ideal hydraulic lift, how do the piston displacements relate? Does it violate energy conservation?

   Incompressibility gives $A_1 x_1 = A_2 x_2$, so $x_2 = x_1\dfrac{A_1}{A_2}$. The large piston moves less. Work in = work out ($F_1x_1=F_2x_2$), so energy is conserved.

   Hint: Equal volume displaced; force gain = distance loss.

10. 10.What is atmospheric pressure at sea level in standard units?

    $P_{atm}\approx 1.013\times10^{5}\,\text{Pa} = 76\,\text{cm}$ of mercury $= 1\,\text{atm} \approx 10.3\,\text{m}$ of water.

    Hint: About $10^5$ Pa or 76 cm Hg.

11. 11.How does a mercury barometer measure atmospheric pressure?

    Atmospheric pressure supports a mercury column: $P_{atm}=\rho_{Hg} g h$. The space above the mercury (Torricelli vacuum) has essentially zero pressure, so $h\approx 76$ cm balances 1 atm.

    Hint: Atmosphere balances a Hg column height.

12. 12.Why is mercury used in a barometer instead of water?

    Mercury's high density ($13.6\times$ water) means the column is short ($\approx 76$ cm vs $\approx 10.3$ m for water). Its low vapour pressure also keeps the vacuum space nearly ideal.

    Hint: High density ⇒ short, practical column.

13. 13.How does an open-tube manometer measure the gauge pressure of a gas?

    The gas pushes the liquid so that $P_{gas}=P_{atm}+\rho g h$ (gas side lower) or $P_{gas}=P_{atm}-\rho g h$. The height difference $h$ gives the gauge pressure $\rho g h$.

    Hint: Height difference of the two arms.

14. 14.Does atmospheric pressure at the barometer reading change if the tube is tilted?

    No. The vertical height of the mercury column stays $76$ cm; only the length along the tilted tube increases. Pressure depends on vertical height, not tube length.

    Hint: Vertical height is what matters.

15. 15.State Archimedes' principle.

    A body fully or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces: $F_B = \rho_{fluid}\, V_{disp}\, g$.

    Hint: Upthrust = weight of displaced fluid.

16. 16.What is the physical origin of the buoyant force?

    It arises from the pressure difference between the bottom and top of the immersed body. Since pressure increases with depth, the upward force on the lower surface exceeds the downward force on the top.

    Hint: Deeper ⇒ higher pressure ⇒ net upthrust.

17. 17.Where does the buoyant force effectively act?

    At the centre of buoyancy — the centroid of the displaced fluid volume. This generally differs from the body's centre of gravity.

    Hint: Centroid of displaced liquid.

18. 18.Give the condition for a body to float in equilibrium.

    Weight equals buoyant force: $\rho_{body} V_{body}\, g = \rho_{fluid} V_{disp}\, g$. For floating, $\rho_{body}<\rho_{fluid}$ and $\dfrac{V_{disp}}{V_{body}}=\dfrac{\rho_{body}}{\rho_{fluid}}$.

    Hint: Weight = upthrust; density ratio gives submerged fraction.

19. 19.What fraction of an iceberg ($\rho=917$ kg/m³) floats above seawater ($\rho=1025$ kg/m³)?

    Submerged fraction $=\dfrac{917}{1025}\approx 0.895$. So about $\mathbf{10.5\%}$ floats above the surface.

    Hint: Above fraction $=1-\rho_{ice}/\rho_{water}$.

20. 20.What is apparent weight of a body immersed in a fluid?

    $W_{app}=W_{real}-F_B = \rho_{body}Vg - \rho_{fluid}Vg$ (fully submerged). The loss of weight equals the weight of displaced fluid.

    Hint: Real weight minus upthrust.

21. 21.Define metacentre and the stability condition for a floating body.

    The metacentre $M$ is where the vertical line through the new centre of buoyancy meets the original vertical axis when tilted. Stable float requires $M$ above the centre of gravity $G$.

    Hint: $M$ above $G$ ⇒ restoring torque.

22. 22.A block floats in water. If some ice on top melts, or the container is on the moon, how does buoyancy change conceptually?

    Buoyancy $=\rho V_{disp} g$ depends on $g$ and displaced volume. On the moon both weight and upthrust scale with $g$, so floating equilibrium (density ratio) is unchanged; the submerged fraction stays the same.

    Hint: Float fraction depends on density ratio, not $g$.

23. 23.Define an ideal fluid used in Bernoulli flow.

    An ideal fluid is incompressible (constant density) and non-viscous (no internal friction). Its flow is assumed steady and irrotational (streamline).

    Hint: Incompressible + non-viscous.

24. 24.Distinguish streamline (laminar) flow from turbulent flow.

    In streamline flow every fluid particle passing a point follows the same smooth path; velocity at each point is steady. In turbulent flow the motion is chaotic with eddies, and streamlines break up above a critical speed.

    Hint: Orderly layers vs chaotic eddies.

25. 25.State the equation of continuity and its basis.

    For incompressible steady flow, $A_1 v_1 = A_2 v_2$, i.e. $Av=\text{const}$. It expresses conservation of mass (volume flow rate is constant).

    Hint: Mass conservation: $Av$ constant.

26. 26.According to continuity, where is fluid speed greatest in a pipe of varying cross-section?

    Where the cross-sectional area is smallest. Since $Av=$ const, $v\propto \dfrac{1}{A}$ — the flow speeds up in a constriction.

    Hint: Narrow ⇒ fast.

27. 27.State Bernoulli's theorem for streamline flow.

    $P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$ along a streamline. It expresses conservation of energy per unit volume for an ideal fluid.

    Hint: Pressure + KE density + PE density = const.

28. 28.What does each term of Bernoulli's equation represent physically?

    $P$ = pressure energy per unit volume; $\tfrac12\rho v^2$ = kinetic energy per unit volume; $\rho g h$ = potential energy per unit volume. Their sum is constant along a streamline.

    Hint: Pressure, kinetic, potential energy densities.

29. 29.List the key assumptions behind Bernoulli's equation.

    Fluid is (1) incompressible, (2) non-viscous, (3) flow is steady and (4) streamline (irrotational), (5) along a single streamline. Viscosity and turbulence violate it.

    Hint: Ideal, steady, streamline flow.

30. 30.State Bernoulli's principle in words (pressure–velocity relation).

    Where the flow speed of a fluid is higher, the pressure is lower (at the same height), and vice versa. Fast flow ⇒ low pressure.

    Hint: Faster flow ⇒ lower pressure.

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