Mechanical Properties of Fluids - Class 11 Part 2

A fluid flows steadily through a horizontal pipe of constant cross-sectional area and length $L$. The pipe rotates with a constant angular velocity $\omega$ about a vertical axis passing through its entrance ($r=0$). Due to an externally maintained temperature profile, the density of the fluid at a radial distance $r$ from the axis is strictly given by $\rho(r) = \rho_0 \left(1 - \frac{2r}{3L}\right)$, where $\rho_0$ is the density at the entrance. Both ends of the pipe are open to the atmosphere. Find the exit velocity $v_L$ of the fluid at $r=L$.

A saturated liquid of density $\rho_0$ and specific heat capacity $c$ enters a horizontal, thermally insulated nozzle at pressure $P_0$ and boiling temperature $T_0$ with negligible velocity. As the fluid accelerates, its pressure drops to $P$, causing a small mass fraction to vaporize. Assume the flow is steady and isentropic, and the vapor density $\rho_v$ is constant. Which of the following expressions gives the velocity $v$ of the mixture at pressure $P$? (Assume the specific latent heat of vaporization $L$ is constant, and the mass fraction of vapor remains small).

Water of density $\rho = 1000 \text{ kg/m}^3$ flows steadily from a large pressurized tank into a horizontal pipe. The total absolute pressure in the tank is known exactly to be $P_0 = 5.0 \times 10^5 \text{ Pa}$. The flow velocity in the pipe is measured as $v = 20 \text{ m/s}$ with a maximum relative error of $3.0\%$. Assuming the density is exact and the fluid is ideal, what is the maximum percentage error in the calculated static pressure $P$ within the pipe?

A rigid, completely sealed rectangular tank of length $L$ and height $H$ is entirely filled with an ideal incompressible liquid of density $\rho$. The tank is given a constant horizontal acceleration $a$ to the right. While accelerating, two small holes are simultaneously opened: one at the top-left corner (rear-top) and another at the bottom-right corner (front-bottom). The ambient atmospheric pressure is $P_0$. Assuming $gH > aL$, what is the velocity of efflux of the liquid from the bottom-right hole relative to the tank?

A large cylindrical tank of cross-sectional area $A$ contains water of density $\rho$. A block of ice of density $\eta \rho$ (where $\eta < 1$) is anchored to the bottom of the tank so that it is completely submerged. A small hole of area $a$ is opened at the bottom of the tank, and water flows out. Simultaneously, a heater supplies heat at a constant rate $P$, which is entirely used to melt the ice. The latent heat of fusion of ice is $L$. If the instantaneous height of the water level is $h$, what is the magnitude of the rate of fall of the water level, $\left|\frac{dh}{dt}\right|$?

A horizontal Venturi tube has a wide section of cross-sectional area $A_1$ and a narrow throat of area $A_2$. An ideal, incompressible fluid of density $\rho$ flows through the tube with a constant volume flow rate $Q$. A rectangular bypass channel of width $w$ and height $d$ connects the wide section to the throat. A rigid dielectric slab of dielectric constant $K$ fits perfectly inside this channel and can slide frictionlessly. The top and bottom walls of the channel act as parallel capacitor plates connected to a constant DC voltage $V$. The slab is partially inserted into the capacitor. The fluid pressure difference exerts a force on the slab that is perfectly balanced by the electrostatic force. Ignoring fringe effects, what is the voltage $V$ required to keep the slab in equilibrium?

An incompressible, electrically conducting fluid of density $\rho$ and electrical conductivity $\sigma$ flows steadily along the $+x$-direction through a horizontal Venturi tube of length $L$. The cross-sectional area of the tube decreases exponentially as $A(x) = A_0 e^{-x/L}$ from $x=0$ to $x=L$. The fluid velocity at the inlet ($x=0$) is $v_0$. A non-uniform transverse magnetic field $\vec{B}(x) = B_0 e^{-x/2L} \hat{k}$ is applied across the tube. The side walls of the tube are electrically short-circuited such that the induced electric field is zero. Assuming one-dimensional inviscid flow, what is the pressure drop $\Delta P = P(0) - P(L)$ across the tube?

A dilute gas of constant macroscopic density $\rho$ flows steadily through a horizontal Venturi tube. The inlet has a cross-sectional area $A$ and the throat has a cross-sectional area $A/2$. The measured pressure drop between the inlet and the throat is $\Delta P$. Inside the throat, a microscopic flat disc of area $a$ ($a \ll A$) is held stationary, perpendicular to the flow. Because the gas is highly dilute (Knudsen number $\gg 1$), it interacts with the disc locally as a stream of discrete, non-interacting particles. Assuming these gas particles collide elastically with the disc, what is the average force exerted by the gas on the disc?

A solid sphere of radius $R$ and density $\rho_s$ falls from rest under gravity in a fluid of density $\rho_f$ and viscosity $\eta$. Due to strong molecular cohesion, the fluid does not obey Stokes' law. Instead, a boundary layer of thickness $\delta$ forms around the sphere, where $\delta = R \left(\frac{v_0}{v}\right)$, with $v$ being the instantaneous velocity of the sphere and $v_0$ a characteristic fluid velocity constant. The velocity profile of the fluid within this boundary layer, from the sphere's surface ($y=0$) to the edge of the layer ($y=\delta$), is given by $u(y) = v\left(1 - \frac{y}{\delta}\right)^2$. Assuming the drag force is entirely due to the viscous shear stress at the sphere's surface, what is the terminal velocity $v_T$ of the sphere?

A grounded spherical conducting liquid drop of radius $R$ and surface tension $T$ is situated at the center of a rigid, isolated spherical conducting shell of radius $kR$ ($k > 1$). The outer shell carries a total charge $Q$. The space between them is vacuum. Assuming the effect of the grounding wire is negligible, what is the excess pressure $\Delta P$ inside the liquid drop?

A theoretical model of evaporation proposes that the effective cohesive 'friction' force resisting a molecule's escape from a liquid surface is given by $F_c = \alpha v^4$, where $v$ is the molecule's speed and $\alpha$ is a positive constant. The surface energy density $\gamma$ (in $\text{J/m}^2$) is defined as the product of the surface number density $n_s$ and the true average work done against $F_c$ over a characteristic escape distance $d$. The molecules in the liquid follow a 3D Maxwell-Boltzmann velocity distribution at temperature $T$.

A student incorrectly calculates the surface energy as $\gamma'$ by assuming all molecules escape with exactly the root-mean-square speed $v_{\text{rms}}$.

Find the ratio $\gamma / \gamma'$. Furthermore, if the measured temperature $T$ has a maximum relative error of $1.5\%$ and the molecular mass $m$ has a maximum relative error of $2.0\%$, what is the maximum percentage error in the true surface energy $\gamma$?

In a theoretical model of a fluid, the molecular cohesive force field near the liquid-gas interface is anisotropic. The interface is parallel to the $xy$-plane at $z = 12L$. For a molecule at position $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ (where $z=0$ represents the deep bulk), the cohesive force is directed entirely perpendicular to the interface and is given by $\vec{F}_c = - \frac{F_0 z^2}{L^2} \hat{k}$ An external agent quasi-statically moves a molecule from the origin $(0,0,0)$ to a point $(3L, 4L, 12L)$ on the liquid surface. What is the work done by the external agent?

A spherical liquid drop of radius $R$ and surface tension $T$ is placed in a vacuum. It is given a total charge $Q$ which is uniformly distributed over its surface. If the outward electrostatic pressure exactly balances the inward pressure due to surface tension, what is the magnitude of the charge $Q$? (Assume $\epsilon_0$ is the permittivity of free space)

A spherical soap bubble of initial radius $R_0$ contains a monoatomic ideal gas. The initial external atmospheric pressure is $P_0 = \frac{12 S}{R_0}$, where $S$ is the surface tension of the soap solution. The system is initially in thermal and mechanical equilibrium. The external pressure is then slowly reduced to zero, allowing the bubble to expand reversibly and adiabatically to a new equilibrium radius $R$. Calculate the work done against the external atmospheric pressure during this expansion.

A massive, isolated spherical shell of an incompressible liquid has an inner radius $R$ and an outer radius $2R$. The liquid has a uniform density $\rho$ and surface tension $S$. The shell encloses a concentric spherical cavity containing a massless gas. The external region outside the shell is a vacuum ($P_{out} = 0$). Assuming the system is in mechanical equilibrium under its own gravitational self-interaction and surface tension, what is the pressure $P_{in}$ of the gas inside the cavity? (Here $G$ is the universal gravitational constant)

A large open cylindrical tank is filled with an ideal liquid of density $\rho$ and surface tension $\gamma$. A narrow horizontal capillary tube of radius $r$ is attached to the side of the tank at a depth $h$ below the free surface of the liquid. The liquid flows out steadily through the capillary into the atmosphere, where the atmospheric pressure is $P_0$. Assuming the liquid forms a spherical meniscus at the exit of the capillary characterized by a contact angle $\theta$, what is the steady efflux velocity $v$ of the liquid? (Assume $r$ is small enough that gravitational distortion of the meniscus is negligible).

A vessel consisting of two large parallel vertical plates separated by a distance $2r$ contains a liquid of density $\rho$ and surface tension $T$. The contact angle of the liquid with the plates is exactly $90^\circ$. At $t=0$, the vessel is subjected to a horizontal jerk $j$ (so its acceleration is $a(t) = jt$). Assuming the liquid surface adjusts quasi-statically and its slope remains small enough that the surface curvature is well-approximated by $\frac{d^2y}{dx^2}$, find the magnitude of the difference in liquid levels between the two plates at time $t$.

A capillary tube of radius $r$ is kept vertically in a large liquid bath. The liquid and the tube are initially at temperature $T_0$ and cool down to the ambient temperature $T_a$ according to Newton's law of cooling with a cooling constant $k$. The surface tension $\gamma$ and contact angle $\theta$ depend on temperature such that the quasi-static equilibrium height of the liquid column is $h_{eq}(T) = h_a - \beta(T - T_a)$, where $h_a$ is the equilibrium height at $T_a$ and $\beta$ is a positive constant. Assuming the mass of the liquid column remains approximately constant as $m = \pi r^2 \rho h_a$ (where $\rho$ is the liquid density) and neglecting viscous damping, the actual height of the liquid column can be expressed as $h(t) = h_{osc}(t) + h_{non-osc}(t)$, where $h_{osc}(t)$ is the oscillatory component. Which of the following represents $h_{non-osc}(t)$?

A solid sphere of radius $r$ and uniform density $\rho$ is released from rest at the surface ($y=0$) of a liquid column. The liquid has a depth-dependent density $\sigma(y) = \sigma_0(1 + \beta y)$ (where $\rho > \sigma_0$) and a depth-dependent viscosity $\eta(y) = \eta_0(1 + \alpha y)$. The sphere is observed to have a velocity $v$ at a depth $y=H$. Which of the following expressions gives the magnitude of the work done by the viscous drag force on the sphere during its descent to depth $H$?

A large open container filled with a viscous liquid of density $\rho_L$ and coefficient of viscosity $\eta$ moves horizontally with a constant acceleration $a = \sqrt{3}g$. A small solid sphere of radius $r$ and density $\rho_S$ (where $\rho_S > \rho_L$) is released from rest relative to the liquid. After a long time, the sphere reaches a steady terminal velocity $\vec{v}_t$ relative to the container. What is the magnitude of this terminal velocity?

A spherical region of radius $R$ contains a viscous gas of viscosity $\eta$ and a fixed spherically symmetric volume charge density $\rho(r) = \rho_0 \frac{r}{R}$, where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. A small oil droplet of charge $q$ and radius $a$ is released from $r = R/2$ and moves radially outward. Assuming the mass of the droplet is negligible so that it instantaneously reaches its local terminal velocity, what is the time taken by the droplet to reach the boundary of the region ($r = R$)? (Ignore gravity)

A solid sphere of initial radius $r_0$ and density $\rho_s$ is falling in a viscous liquid of density $\rho_l$ (where $\rho_s > \rho_l$). The system is slowly heated such that the sphere's radius expands as $r(t) = r_0(1 + \alpha t)$ and the liquid's viscosity decreases as $\eta(t) = \eta_0(1 - \beta t)$, where $\alpha$ and $\beta$ are small positive constants. Assuming the sphere continuously maintains its quasi-steady terminal velocity at each instant, what is the acceleration of the sphere at $t = 0$?

A uniform solid sphere of mass $m$ and radius $R$ is initially at rest in a large tank of viscous fluid. At $t=0$, an impulsive torque delivers an angular impulse $J$ to the sphere about its central axis. The fluid exerts a viscous retarding torque $\tau = -k\omega$, where $\omega$ is the instantaneous angular velocity and $k$ is a positive constant. What is the total angular displacement $\Delta \theta$ of the sphere before it comes to rest, and the total heat energy $H$ dissipated by the viscous fluid?

A block of mass $m = 1 \text{ kg}$ is attached to a horizontal ideal spring of spring constant $k = 100 \text{ N/m}$. The block oscillates in a fluid where the drag force transitions from linear (Stokes regime) to quadratic (Newtonian turbulent regime) depending on the velocity $v$, modeled as $F_d = -b v - c v |v|$, where $b = \frac{0.4}{\pi} \text{ kg/s}$ and $c = 0.03 \text{ kg/m}$. The block is released from rest at a displacement $A = 1 \text{ m}$. Assuming the energy dissipated per cycle is small compared to the total energy (weak damping approximation), what is the approximate total mechanical energy dissipated by the fluid drag during the first quarter cycle (from $x = A$ to $x = 0$)?

A conducting incompressible fluid of density $\rho$, dynamic viscosity $\eta$, and electrical conductivity $\sigma$ flows with an average velocity $v$ through a non-conducting cylindrical pipe of diameter $d$. A uniform transverse magnetic field $B$ is applied. The induced eddy currents generate a local Lorentz body force. A student incorrectly models the eddy currents as closed magnetic dipoles and concludes that since the field is uniform, the net magnetic force is zero. In reality, the local Lorentz force acts as an additional velocity-dependent drag. Assuming the inertial force per unit volume is $f_i = \rho \frac{v^2}{d}$, the standard viscous force per unit volume is $f_v = \eta \frac{v}{d^2}$, and the magnetic drag force per unit volume is $f_m = \sigma v B^2$, the flow transitions to turbulence when the effective Reynolds number (ratio of inertial force to total drag force) reaches a critical value $Re_0$. What is the correct critical velocity $v_c$ for the onset of turbulence?

A two-dimensional open container has a parabolic cross-section given by $y = c x^2$, where $y$ is the vertical axis and $x$ is the horizontal axis. The container is filled with an incompressible liquid of density $\rho$ up to a height $H$ (when at rest). The container is then given a constant horizontal acceleration $a$ in the $+x$ direction. Assuming the liquid does not spill, what is the steady-state gauge pressure at the lowest point of the container (the origin)?

A closed conical container of height $H$ and base area $A$ is placed with its flat base on a horizontal surface and its vertex pointing vertically upwards. The container is completely filled with a non-uniform incompressible fluid whose density varies with depth $y$ (measured downwards from the vertex) as $\rho(y) = \rho_0 \left(1 + \frac{y}{H}\right)$, where $\rho_0$ is a positive constant. Let $g$ be the acceleration due to gravity. Neglecting atmospheric pressure, what is the magnitude of the net vertical force exerted by the curved inner walls of the container on the fluid?

To analyze the Hydrostatic Paradox, a physicist models the gauge pressure $P$ of a fluid as the energy density of a fictitious electric field $\vec{E} = -E(z)\hat{k}$, such that $P(z) = \frac{1}{2}\epsilon_0 |\vec{E}|^2$. A conical container with a flat horizontal base of radius $R$ at $z=0$ and apex at $z=H$ is completely filled with an incompressible fluid of uniform density $\rho$. The container is in a uniform downward gravitational field $g$, and the pressure at the apex is zero. By applying Gauss's Law to the fictitious field $\vec{E}$, determine the total fictitious charge $Q_{enc}$ enclosed within the fluid volume.

A fixed rigid container has a cross-sectional area that decreases with height. The base of the container is a massless, frictionless piston of area $A_0$. The container contains an ideal, incompressible fluid of density $\rho$. The piston is pushed upwards, moving with a constant vertical velocity $v_0$. At a certain instant, the fluid's free surface is at a height $H$ above the piston, where the container's cross-sectional area is $A_1 = \frac{A_0}{2}$. Atmospheric pressure is $P_0$ and the acceleration due to gravity is $g$. What is the external upward force $F$ required on the piston at this instant?

A cylindrical container of height $H$ is filled with a liquid of density $\rho$ and optical absorption coefficient $\alpha$. A uniform light beam of intensity $I_0$ is incident normally on the entire free surface of the liquid. Assuming the refractive index of the liquid is $n=1$ and the free surface does not reflect any light, what is the gauge pressure of the liquid at a depth $H$? (Here $c$ is the speed of light in vacuum, and assume the momentum of a photon is $E/c$).

A non-cylindrical tank has a height-dependent cross-sectional area $A(h) = A_0 \sqrt{\frac{h}{H}}$, where $H$ is the initial height of the incompressible liquid (density $\rho$) and $h$ is the instantaneous height. A small hole of area $a$ ($a \ll A_0$) at the bottom is open to the atmosphere at a constant pressure $P_0$. The tank is sealed at the top, and a computer-controlled valve maintains the gas pressure above the liquid at $P(t) = P_0 + \rho g (H - h) + \frac{1}{2}\rho \gamma t$, where $\gamma = \frac{a}{A_0} (2g)^{3/2} \sqrt{H}$ and $g$ is the acceleration due to gravity. Assuming quasi-steady flow and negligible gas density, find the time $T$ taken to completely drain the tank.

A very large open tank is filled with an incompressible liquid of uniform mass density $\rho$ and a uniform volume charge density $\rho_e$. An infinite, thin, non-conducting horizontal sheet carrying a uniform surface charge density $+\sigma$ is held fixed at a depth $h$ below the free surface of the liquid. The atmospheric pressure is $P_0$ and the permittivity of the liquid is $\epsilon$. Assuming the electric field produced by the liquid's own volume charge is negligible, what is the absolute pressure $P$ at a depth $d$ ($d > h$) below the free surface?

A large open cylindrical tank of height $H$ is filled with a liquid. The temperature of the liquid varies linearly with depth $y$ from the surface, being $T_0$ at the surface ($y=0$) and $T_0 + \Delta T$ at the bottom ($y=H$). The density of the liquid varies with temperature as $\rho(T) = \rho_0 [1 - \gamma (T - T_0)]$, where $\rho_0$ is the density at $T_0$ and $\gamma$ is the coefficient of volume expansion. A small hole is punched at the bottom of the tank. Assuming the fluid in the tank is practically at rest initially, what is the initial velocity of efflux $v$ from the hole? (Atmospheric pressure is $P_0$)

A spherical planet of radius $R$ consists of a solid core of radius $R/3$ and uniform density $4\rho$, surrounded by an incompressible fluid of uniform density $\rho$ up to the surface $r=R$. The pressure at the surface of the planet is zero. What is the hydrostatic pressure at the surface of the solid core? (Assume $G$ is the universal gravitational constant)

A closed vertical cylinder of height $H$ is filled with a mixture of two ideal gases, A and B, having molar masses $M$ and $2M$ respectively. The system is in hydrostatic equilibrium in a uniform downward gravitational field $g$. The temperature of the gas mixture varies with height $z$ from the base as $T(z) = T_0(1 + \alpha z)$, where $T_0$ and $\alpha$ are positive constants. If the partial pressures of gases A and B at the base ($z=0$) are both equal to $P_0$, which of the following expressions gives the total pressure of the gas mixture at height $z=H$? (Here $R$ is the universal gas constant).

A viscous, incompressible fluid of viscosity $\eta$ flows steadily through a horizontal pipe consisting of two cylindrical sections in series. The first section has radius $R$ and length $L$, while the second section has radius $R/2$ and length $L$. In the first section, the fully developed laminar velocity profile is given by $v_1(r) = v_0 \left(1 - \frac{r^2}{R^2}\right)$, where $r$ is the radial distance from the central axis. Assuming fully developed laminar flow is maintained in the second section as well, what is the total viscous drag force exerted by the fluid on the cylindrical walls of the entire pipe?

A long thin-walled elastic cylindrical tube of unstretched radius $R$ and thickness $h$ ($h \ll R$) is made of a material with Young's modulus $Y$. The tube is completely filled with a compressible fluid of density $\rho$ and bulk modulus $B$. A piston at one end suddenly pushes inward, generating a pressure pulse that propagates along the tube. Assuming the tube expands only radially (longitudinal strain is negligible), what is the speed $v$ of the pressure wavefront propagating through the fluid?

A solid dielectric block of mass $M$, height $L$, base area $A$, and dielectric constant $K$ is partially submerged in a non-conducting, unpolarized liquid of density $\rho$. The liquid surface lies at $y=0$, with the $y$-axis pointing vertically downwards. An external non-uniform electric field $\vec{E} = E_0 \sqrt{\frac{y}{L}} \hat{j}$ exists in the region $y \ge 0$, while $\vec{E} = \vec{0}$ for $y < 0$. If the block is in vertical equilibrium, find its submerged depth $x$.

A solid cubical block of side length $L = 2\text{ m}$ and uniform density $\rho_b = 1100\text{ kg/m}^3$ is released from rest such that its lower face just touches the surface of an inviscid liquid. The density of the liquid varies with depth $y$ as $\rho(y) = \rho_0 + ky$, where $\rho_0 = 1000\text{ kg/m}^3$ and $k = 600\text{ kg/m}^4$. Assuming the liquid is deep enough and the block does not tumble, what is the maximum depth $H$ reached by the bottom face of the block? (Take $g = 10\text{ m/s}^2$)

A solid non-magnetic cylinder of mass $M$ and cross-sectional area $A$ floats vertically in a non-magnetic liquid of density $\rho$. A tightly wound circular coil of $N$ turns and radius $R$ ($R \ll \sqrt{A}$) is fixed at the base of the cylinder. The system is placed in an external, axially symmetric magnetic field whose vertical component on the axis is given by $B_z = B_0(1 - \alpha z)$, where $z$ is measured upwards from the undisturbed liquid surface. A steady current $I$ is passed through the coil such that its magnetic moment is directed upwards. Assuming the liquid surface remains flat and ignoring the magnetic field produced by the coil itself, what is the additional depth $\Delta h$ to which the cylinder sinks?

A solid sphere of external radius $R$ and a concentric internal cavity of radius $R_c$ is completely submerged in a liquid of constant density $\rho_L$. The sphere is held in equilibrium by a thin vertical wire. The interatomic potential of the solid's material near its equilibrium atomic spacing $r_0$ is modeled as $U(r) = U_0 + \frac{1}{2}k(r-r_0)^2 - b(r-r_0)^3$, where $k$ and $b$ are positive constants and $r$ is the interatomic distance. If the temperature of the system is increased by a small amount $\Delta T$, which of the following represents the change in the buoyant force $\Delta F_B$ acting on the sphere? (Assume the liquid's density is unaffected by temperature, $k_B$ is the Boltzmann constant, and $g$ is the acceleration due to gravity.)

A cylindrical container filled with a fluid rotates with a constant angular velocity $\omega$ about its vertical axis. The $z$-axis is taken vertically downwards along the axis of rotation, and $r$ is the radial distance from the axis. Due to a specific thermal and solute gradient, the density of the fluid in the steady rotating frame is maintained as $\rho(r,z) = \rho_0 (1 + \alpha z + \beta r^2)$, where $\alpha$ and $\beta$ are positive constants. The acceleration due to gravity is $g$ downwards. For this steady-state relative equilibrium to be physically possible, which of the following conditions must hold?

A rectangular tank of length $L$ and width $b$ is partially filled with an ideal, incompressible liquid of density $\rho$ to a depth $H$. The tank is forced to oscillate horizontally such that its position is given by $x(t) = A \sin(\omega t)$. Assume the frequency $\omega$ is low enough that the free surface remains planar (quasi-static approximation) and the liquid does not spill. Which of the following expressions represents the maximum rate of change of the gravitational potential energy of the liquid?

An open rectangular tank containing an ideal liquid slides down a fixed rough inclined plane of inclination $\theta$. The tank starts from rest at $x = 0$. The coefficient of kinetic friction between the tank and the incline varies with the distance $x$ moved down the incline as $\mu = kx$, where $k$ is a positive constant. Assuming the tank is sufficiently long and deep so that the liquid does not spill, what is the maximum angle $\phi$ that the free surface of the liquid makes with the inclined plane during the downward motion before the tank momentarily comes to rest?

A horizontal Venturi meter is mounted on a cart that moves with a constant horizontal acceleration $a$ in the direction of the fluid flow. The fluid has density $\rho$. The cross-sectional areas of the wide and narrow sections of the meter are $A$ and $a_0$, respectively. A U-tube manometer containing a liquid of density $\rho_m$ ($\rho_m > \rho$) is connected between the wide and narrow sections. The horizontal distance between the two tapping points is $L$. Assume the flow rate is large enough such that the liquid level in the manometer arm connected to the wide section is lower than that in the other arm by a height $h$. Which of the following expressions gives the volume flow rate $Q$ of the fluid?

A thin-walled hollow cylinder of mass $M$, radius $R$, and length $L$ is completely filled with an inviscid, incompressible fluid of density $\rho$. The total mass of the fluid is $m$. The cylinder rests on an inclined plane of angle $\theta$. The coefficient of kinetic friction between the cylinder and the plane is $\mu_k$. A horizontal impulsive force $J$ is applied at the center of mass of the cylinder, directed such that it pushes the cylinder into the incline. Assume $\cot\theta > \mu_k$ so that the cylinder slips down the incline during the impulse. Let $\Delta \Pi$ be the magnitude of the difference in pressure impulse between the point in the fluid furthest down the incline and the point furthest up the incline. Let $\Delta \omega$ be the angular velocity acquired by the cylinder just after the impulse. Which of the following expressions gives the ratio $\frac{\Delta \Pi}{R \Delta \omega}$?

An ideal, inviscid fluid flows steadily through a horizontal pipe of constant cross-sectional area $A$. The fluid enters the pipe at $x=0$ with velocity $v_0$, temperature $T_0$, and density $ho_0$. As the fluid flows, it is heated such that its temperature increases linearly, reaching $T_0 + \Delta T$ at $x=L$. The coefficient of volume expansion of the fluid is $\gamma$ (assume $\gamma \Delta T \ll 1$). Assuming the density variation is solely due to the temperature change, which of the following expressions correctly represents the pressure drop $\Delta P = P(0) - P(L)$ across the pipe?

A rigid pipe of length $L = 5 \text{ m}$ is fixed such that it slopes upward and to the right at an angle $\theta = 37^\circ$ to the horizontal. The entire setup accelerates horizontally to the right with a constant acceleration $a = 5 \text{ m/s}^2$. An ideal incompressible fluid of density $\rho = 1000 \text{ kg/m}^3$ flows steadily up the pipe. At the lower end (point 1), the cross-sectional area is $A_1 = 20 \text{ cm}^2$ and the fluid velocity relative to the pipe is $v_1 = 2 \text{ m/s}$. At the upper end (point 2), the cross-sectional area is $A_2 = 10 \text{ cm}^2$. Assuming the flow is steady in the accelerating frame, what is the pressure difference $P_1 - P_2$ between the lower and upper ends? (Take $g = 10 \text{ m/s}^2$, $\sin 37^\circ = 3/5$)

A snowplow moves at a constant speed $v$ directly up a rough inclined plane of angle $\theta$. The plane is initially covered with a uniform layer of stationary snow of density $\rho$ and cross-sectional area $A$. The plow collects the snow, which forms a plug of mass $M$ that slides up the incline at speed $v$ just in front of the plow. The coefficient of kinetic friction between the snow plug and the incline is $\mu$. What is the total force $F$ exerted by the snowplow on the snow plug at the instant its mass is $M$?

An ideal gas of molar mass $M$ and adiabatic exponent $\gamma$ is kept in a large insulated stagnation tank at a constant pressure $P_0$ and temperature $T_0$. The gas flows out steadily through a small horizontal converging nozzle of exit area $A$ into the atmosphere at pressure $P$ (assume $P$ is high enough such that the flow is not choked). Which of the following expressions correctly represents the mass flow rate $\dot{m}$ of the gas?