Thermal Properties of Matter - Class 11 Part 2

A faulty thermometer reads $5^\circ\text{X}$ at the melting point of ice and $95^\circ\text{X}$ at the boiling point of water at standard atmospheric pressure. A uniform metal rod is perfectly constrained between two rigid, immovable walls. The rod is heated, and the faulty thermometer records its initial and final temperatures as $14^\circ\text{X}$ and $86^\circ\text{X}$, respectively. The rod has a constant coefficient of linear thermal expansion $\alpha = 2 \times 10^{-5} \text{ } ^\circ\text{C}^{-1}$. Its Young's modulus varies with the true Celsius temperature $T$ as $Y(T) = Y_0(1 - \beta T)$, where $Y_0 = 2 \times 10^{11} \text{ N/m}^2$ and $\beta = 5 \times 10^{-3} \text{ } ^\circ\text{C}^{-1}$. Assuming the rod does not buckle, what is the magnitude of the thermal stress developed in the rod?

A thermistor has a temperature-dependent resistance given by $R(T) = R_0 e^{\beta/T}$, where $T$ is the absolute temperature in Kelvin. It is given that $R_0 = 10\ \Omega$ and $\beta = 300 \ln(3)\ \text{K}$. The thermistor is connected in series with an inductor of inductance $L = 0.4\ \text{H}$ and an AC voltage source $V(t) = 100 \sin(100t)\ \text{V}$. The temperature is measured using a faulty linear scale $^\circ X$, on which the freezing and boiling points of water are $10^\circ X$ and $110^\circ X$, respectively. In steady state, the thermistor reaches a constant operating temperature measured as $37^\circ X$, while the surrounding environment is at $7^\circ X$. Assuming Newton's law of cooling holds, what is the thermal dissipation constant $K$ of the thermistor?

An ideal diatomic gas ($\gamma = 1.4$) is initially at a temperature of $27^\circ\text{C}$. It is compressed adiabatically to $\frac{1}{32}$ of its original volume. A faulty thermometer is used to measure the temperature of the gas. This thermometer exhibits a linear response, reading $10^\circ\text{X}$ at the melting point of ice and $160^\circ\text{X}$ at the boiling point of water at standard atmospheric pressure. What is the final temperature of the gas as measured by this faulty thermometer? (Assume $0^\circ\text{C} = 273\text{ K}$)

A damped harmonic oscillator of mass $m = 0.2\text{ kg}$ is submerged in a viscous fluid. The damping force is given by $F_d = -b v$, where the damping coefficient $b$ depends on the absolute temperature $T$ (in Kelvin) of the fluid as $b = \alpha T$, with $\alpha = 4 \times 10^{-3}\text{ kg s}^{-1}\text{ K}^{-1}$. The temperature of the fluid is measured using a faulty linear thermometer scale $^\circ X$. On this scale, the freezing point of water is $20^\circ X$ and the boiling point of water is $120^\circ X$ (at standard atmospheric pressure). During the experiment, the faulty thermometer reads $47^\circ X$, and the damped angular frequency of the oscillator is measured to be $\omega_d = 4\text{ rad s}^{-1}$. What is the undamped natural angular frequency $\omega_0$ of the oscillator?

Two linear thermometers, A and B, have faulty calibrations. Thermometer A reads the freezing and boiling points of water as $10^\circ \text{A}$ and $130^\circ \text{A}$, respectively. Thermometer B reads the same points as $-20^\circ \text{B}$ and $140^\circ \text{B}$, respectively.

$2 \text{ moles}$ of a monatomic ideal gas are initially at a temperature of $106^\circ \text{A}$. $3 \text{ moles}$ of a diatomic ideal gas are initially at a temperature of $-4^\circ \text{B}$. The two gases are mixed in a rigid, adiabatic container of fixed volume $V = 8.314 \text{ m}^3$.

Assuming no chemical reaction occurs, what is the final equilibrium pressure of the gas mixture? (Take the universal gas constant $R = 8.314 \text{ J mol}^{-1} \text{K}^{-1}$ and absolute zero as $-273^\circ \text{C}$.)

A thick uncharged conducting spherical shell has an inner radius $R_1$ and outer radius $R_2$. The spherical cavity ($0 \le r < R_1$) is completely filled with a uniformly charged compressible fluid of total charge $Q$ which always fills the cavity. The shell is made of a material whose coefficient of linear thermal expansion varies radially as $\alpha(r) = \frac{c}{r^2}$ for $R_1 \le r \le R_2$, where $c$ is a positive constant. The outer surface of the shell ($r = R_2$) is mechanically fixed such that it cannot move. The system is heated, increasing its temperature at a constant rate $\frac{dT}{dt} = \beta$. Assuming the radial displacement $u(r)$ of the shell material is purely kinematic and governed by $\frac{\partial u}{\partial r} = \alpha(r) \Delta T$, find the rate of change of the electric field magnitude $\frac{\partial E}{\partial t}$ at a fixed distance $r$ ($r < R_1$) from the center at $t=0$.

A cylindrical tank of constant cross-sectional area $A_1$ contains an ideal, incompressible liquid of density $\rho$. At the bottom of the tank is a circular orifice made of a material with a coefficient of linear thermal expansion $\alpha$. The tank walls have zero thermal expansion. Initially ($t=0$), the liquid is at a height $H$ above the orifice, and the orifice has an area $A_0$ such that $A_1 = 2A_0$. The system is heated such that the temperature of the orifice increases at a constant rate $\frac{dT}{dt} = c$. Assuming the liquid density remains constant, find the initial rate of change of the efflux velocity of the liquid, $\left. \frac{dv_2}{dt} \right|_{t=0}$.

An unconstrained isotropic solid block contains a spherical cavity of radius $R$ centered at the origin $(0,0,0)$. The block is subjected to a steady-state temperature distribution $T(x,y,z) = T_0 + \beta x$, where $T_0$ and $\beta$ are positive constants. The rigid body motion of the block is constrained such that the theoretical displacement and rotation at the origin are zero. If $\alpha$ is the coefficient of linear thermal expansion of the material, what is the displacement vector of the point on the cavity boundary originally located at $(0, R, 0)$? (Assume $\alpha \beta R \ll 1$).

A solid isotropic medium contains a spherical cavity of volume $V_0$. The cavity is filled with one mole of an ideal monatomic gas at an initial temperature $T_0$ and pressure $P_0$. The solid is heated quasi-statically. The cavity volume $V$ responds to both the temperature $T$ of the solid and the internal gas pressure $P$ according to the relation: $V = V_0 \left[ 1 + \gamma (T - T_0) + \frac{P - P_0}{K} \right]$ where $\gamma = \frac{5}{2T_0}$ is the volumetric thermal expansion coefficient and $K = 2P_0$ is an effective elastic stiffness parameter. Assuming the gas and the solid are always in thermal equilibrium, what is the molar heat capacity of the gas at the beginning of the heating process ($T = T_0$)?

A Young's Double Slit Experiment (YDSE) apparatus is constructed entirely from a solid isotropic material with a coefficient of linear thermal expansion $\alpha$. The slits are separated by a distance $d$ and the screen is rigidly attached at a distance $D$ from the slit plane. The apparatus is illuminated by a coherent light source of wavelength $\lambda$ (which remains unaffected by the temperature of the apparatus). Initially, at temperature $T$, the $n^{\text{th}}$ order bright fringe is observed at a distance $y_n$ from the central maximum. If the entire apparatus is uniformly heated to a temperature $T + \Delta T$, what is the shift $\Delta y_n$ in the position of the $n^{\text{th}}$ order bright fringe? (Assume $\alpha \Delta T \ll 1$ and retain leading-order terms in $\alpha \Delta T$)

A solid sphere of density $\rho_1$ at temperature $T_0$ is released from rest at the surface ($y=0$) of a deep, non-viscous liquid column. The temperature of the liquid varies with depth $y$ as $T(y) = T_0 - ky$, where $k$ is a positive constant. The density of the liquid at $T_0$ is $\rho_2$, and it is given that $\rho_1 = \frac{5}{4}\rho_2$. The volume expansion coefficient of the liquid is $\gamma_l = 5\alpha$ and that of the sphere's material is $\gamma_s = 2\alpha$. Assuming $\alpha k y \ll 1$ so that higher-order terms can be neglected, what is the maximum depth $H$ reached by the sphere before it momentarily comes to rest?

A parallel plate capacitor of initial capacitance $C_0$ is connected to a constant voltage source $V$. The space between the plates is completely filled with a non-polar liquid dielectric of initial dielectric constant $K_0$ at temperature $T_0$. The liquid has a volume expansion coefficient $\gamma$. The capacitor plates are held rigidly at a constant separation, and a small reservoir allows the liquid to expand out so the volume between the plates remains constant. Assuming the electric susceptibility of the liquid is directly proportional to its density, what is the magnitude of the change in the electrostatic energy stored in the capacitor when the temperature is increased by a small amount $\Delta T$? (Assume $\gamma \Delta T \ll 1$)

A thermally insulated, solid radioactive sphere of initial radius $R_0$, mass $m$, and specific heat capacity $s$ is placed in a vacuum. At $t=0$, it contains $N_0$ active nuclei with a decay constant $\lambda$. Each decay releases energy $Q$, which is completely absorbed by the sphere. The coefficient of linear thermal expansion of the sphere's material is $\alpha$. Gamma photons emitted from the center of the sphere travel to the surface, undergoing attenuation according to $I = I_0 e^{-\tau}$, where the optical depth $\tau = \mu R$, and the linear attenuation coefficient $\mu$ is directly proportional to the instantaneous density $\rho$ of the sphere. If the initial optical depth is $\tau_0$, which of the following expressions represents the optical depth $\tau(t)$ at time $t$? (Assume $\alpha \Delta T \ll 1$).

A tall cylindrical tank of depth $H$ is completely filled with a liquid. The temperature of the liquid varies linearly with depth $y$ (measured downwards from the free surface) as $T(y) = T_0 + c y$, where $T_0$ is the surface temperature and $c$ is a positive constant. The density of the liquid varies with temperature as $\rho(T) = \frac{\rho_0}{1 + \gamma(T - T_0)}$, where $\rho_0$ is the density at $T_0$ and $\gamma$ is the coefficient of volume expansion. Assuming acceleration due to gravity $g$ is constant and atmospheric pressure is $P_0$, the absolute pressure at the bottom of the tank is:

A block of ice of mass $m$ and density $\rho_i$ is released from rest from a height $H$ above the free surface of a liquid of density $\rho_L$ ($\rho_L < \rho_i$). The liquid has mass $M$ and is contained in a perfectly insulated, rigid, closed cylindrical calorimeter. The space above the liquid is a vacuum. The initial depth of the liquid is $d$. The ice block falls, sinks to the bottom of the calorimeter, and then melts completely. The initial temperature of the liquid is $T_0 > 0^\circ\mathrm{C}$ and the ice is at $0^\circ\mathrm{C}$. The specific heat capacity of the liquid is $s_L$, the specific heat capacity of water is $s_w$, and the latent heat of fusion of ice is $L$. The density of water is $\rho_w$. Assume the liquid and water are immiscible, and the cross-sectional area of the calorimeter is very large such that the depth of the melted water layer is negligible. Which of the following equations correctly represents the energy balance to find the final equilibrium temperature $T_f$ of the mixture?

A nuclear micro-reactor powers a heating system by converting mass into energy. The rate of mass defect in the reactor varies with time as $\frac{dm}{dt} = \lambda t$, where $\lambda = 1.4 \times 10^{-13} \text{ kg/s}^2$. The reactor operates from $t = 0$ to $t = 10 \text{ s}$. The entire energy released is used to heat a mixture of two liquids, A and B, kept in a perfectly insulated container. Liquid A has a mass of $1 \text{ kg}$ and a temperature-dependent specific heat capacity $s_A(T) = \alpha T^2$, where $\alpha = 0.01 \text{ J kg}^{-1} \text{K}^{-3}$. Liquid B has a mass of $2 \text{ kg}$ and a specific heat capacity $s_B(T) = \alpha T^2$. If the initial temperature of the mixture is $100 \text{ K}$, find the final temperature of the mixture. (Take the speed of light $c = 3 \times 10^8 \text{ m/s}$).

A cylindrical heating wire of initial resistance $R_0 = 1 \text{ }\Omega$ at $T_0 = 300 \text{ K}$ is submerged in an insulating vessel containing $m = 1 \text{ kg}$ of a liquid with specific heat capacity $s = 4000 \text{ J kg}^{-1} \text{K}^{-1}$. A constant voltage $V = 20 \text{ V}$ is applied across the wire. The temperature coefficient of resistivity of the wire is $\alpha_\rho = 5 \times 10^{-3} \text{ K}^{-1}$ and its coefficient of linear thermal expansion is $\alpha_L = 1 \times 10^{-3} \text{ K}^{-1}$. Assuming the heat capacity of the wire and the vessel are negligible, and no heat is lost to the surroundings, find the time $t$ required to heat the liquid to $T = 350 \text{ K}$. (Use the approximation $(1+x)^n \approx 1+nx$ for small $x$).

A block of mass $m$ performs simple harmonic motion with amplitude $A$ and angular frequency $\omega$. The block acts as a heat source, generating thermal energy at an instantaneous rate $P(t) = \alpha U(t)$, where $U(t)$ is the instantaneous potential energy of the oscillator (taking $U=0$ at the mean position) and $\alpha$ is a positive constant. This heat is supplied to a mixture of ice and water at $0^{\circ}\mathrm{C}$. However, due to a thermal gating mechanism, heat is transferred to the mixture only when the block's displacement satisfies $|x| \ge A/2$. If the latent heat of fusion of ice is $L$, what is the mass of ice melted in one complete time period of the oscillation?

A spherical region of radius $R$ contains a weakly conducting ice-water mixture at $0^\circ\text{C}$ with uniform electrical conductivity $\sigma_c$ and permittivity $\epsilon_0$. An external mechanism maintains a steady, non-uniform volume charge density $\rho(r) = \rho_0 \frac{r}{R}$ throughout the sphere, where $r$ is the radial distance from the center. The resulting electric field causes Joule heating within the mixture. Assuming no heat is lost to the surroundings and the latent heat of fusion of ice is $L_f$, what is the mass of ice melted in time $t$?

One mole of an ideal monatomic gas is enclosed in a cylinder with a movable piston. Starting at $t=0$, the gas expands and does work on the piston at a constant rate $W_0$. Simultaneously, an internal electric heater supplies heat to the gas at a constant rate $P_0$ (where $P_0 > W_0$). The cylinder walls are poorly insulated, resulting in a time-varying heat loss to the surroundings at a rate $\frac{dQ_{\text{loss}}}{dt} = \alpha t$, where $\alpha$ is a positive constant. If the initial temperature of the gas is $T_0$, what is the maximum temperature reached by the gas during this process? (Here $R$ is the universal gas constant)

A lead bullet of mass $m$, specific heat capacity $c$, surface area $A$, and emissivity $e$ is fired into a large target. The bullet enters the target at $t=0$ with an initial velocity $v_0$ and experiences a constant retarding force $F$. It comes to rest at time $t = t_0$. The initial temperature of the bullet and the constant temperature of the surroundings are both $T_0$. Assuming all the kinetic energy lost by the bullet is continuously converted into heat within the bullet, and it simultaneously radiates heat to the surroundings according to the Stefan-Boltzmann law, which of the following expressions gives the exact net temperature rise $\Delta T = T(t_0) - T_0$ of the bullet? (Assume the bullet does not melt, $T(t)$ is its temperature at time $t$, and $\sigma$ is the Stefan-Boltzmann constant).

A heating coil is submerged in a calorimeter containing a liquid. The coil is connected to a constant voltage source of $V = 20\ \text{V}$. The resistance of the coil at the surrounding temperature $T_s = 25^\circ\text{C}$ is $R_0 = 10\ \Omega$, and its temperature coefficient of resistance is $\alpha = 0.05\ ^\circ\text{C}^{-1}$. The calorimeter loses heat to the surroundings according to Newton's law of cooling, with a heat transfer coefficient $k = 1\ \text{W/}^\circ\text{C}$. Assuming the liquid is continuously stirred and maintains a uniform temperature, what is the steady-state temperature of the liquid?

Two solid blocks, A and B, have constant heat capacities $C$ and $2C$, and initial temperatures $3T_0$ and $2T_0$ respectively. They are placed inside a massive evacuated chamber whose walls are maintained at a constant temperature $T_0$. The blocks are connected by a thermally conducting rod, allowing them to exchange heat with each other. Simultaneously, they lose heat to the chamber walls via non-linear Stefan-Boltzmann radiation. After a long time, the system reaches thermal equilibrium. What is the total change in entropy of the universe?

A heating coil of resistance $R$ and inductance $L$ is immersed in a calorimeter of heat capacity $C$. At $t=0$, the coil is connected to an ideal DC voltage source $V_0$. The calorimeter loses heat to the surroundings (at constant temperature $T_0$) at a rate $k(T - T_0)$, where $T$ is the instantaneous temperature of the calorimeter and $k$ is a positive constant. Assume the electrical time constant of the coil is exactly equal to the thermal time constant of the calorimeter, i.e., $\frac{L}{R} = \frac{C}{k} = \tau$. If the initial temperature of the calorimeter is $T_0$, which of the following expressions correctly gives the temperature difference $\Delta T(t) = T(t) - T_0$ at any time $t > 0$?

A cylindrical vessel of cross-sectional area $A$ is placed in an elevator accelerating upwards with a constant acceleration $a$. The vessel contains a mass $m$ of ice at $0^\circ\text{C}$. A frictionless piston of mass $M$ rests on the ice, exposed to atmospheric pressure $P_0$. Heat is slowly supplied until the ice completely melts into water at $0^\circ\text{C}$. Let $L$ be the increase in internal energy per unit mass during melting. The densities of ice and water are $\rho_1$ and $\rho_2$ respectively ($\rho_2 > \rho_1$). What is the total heat $Q$ supplied to the system? (where $g$ is the acceleration due to gravity)

A microscopic system of initial rest mass $M$ is initially at rest. It undergoes a phase transition where its internal rest energy decreases by $\Delta U$. During the transition, the system expands by a volume $\Delta V$ against a constant external pressure $P$, doing mechanical work $P\Delta V$. The remaining available energy is emitted as a single photon, and the system recoils freely in vacuum. Which of the following gives the exact energy $E_\gamma$ of the emitted photon? (Here $c$ is the speed of light in vacuum).

A thin spherical shell of radius $r$ and thickness $t$ ($t \ll r$) is completely filled with a liquid of mass $M$ and specific volume $v_0$ at its freezing temperature $T_0$. The shell material has Young's modulus $Y$, Poisson's ratio $\nu$, coefficient of linear thermal expansion $\alpha$, and tensile yield stress $\sigma_y$. The system is slowly cooled. Upon freezing, the substance transforms into a solid with a larger specific volume $v_s$ (let $\Delta v = v_s - v_0 > 0$), releasing a latent heat of fusion $L$. The expansion during freezing increases the internal pressure, which in turn lowers the freezing point via the Clausius-Clapeyron relation. Assuming the specific volumes of the liquid and solid phases are independent of pressure and temperature, what fraction $f$ of the liquid has frozen exactly when the shell begins to yield?

A spherical melting ice particle falls through a warm fluid. Due to uniform heat transfer, its radius $r$ decreases at a constant rate $c$ (i.e., $\frac{dr}{dt} = -c$). The particle always falls at its instantaneous terminal velocity. For large radii, the fluid flow is turbulent and the drag force is $F_D = k_1 r^2 v^2$. For small radii, the flow is laminar and the drag force is $F_D = k_2 r v$. The effective weight of the particle (gravity minus buoyancy) is given by $W_{eff} = k_3 r^3$. Assuming the transition between the drag regimes occurs abruptly when the terminal velocities predicted by both models are equal, what is the total distance $H$ the particle falls from an initial radius $r_0$ (where $r_0$ is in the turbulent regime) until it melts completely?

A small solid body of heat capacity $C = 20 \text{ J/K}$ is initially at a temperature $T_0 = 0^\circ\text{C}$. It is connected to two large heat baths maintained at temperatures $T_1 = 100^\circ\text{C}$ and $T_2 = 0^\circ\text{C}$ through two distinct uniform rods of negligible heat capacity. The thermal resistances of the rods are $R_1 = 5 \text{ K/W}$ (connected to $T_1$) and $R_2 = 20 \text{ K/W}$ (connected to $T_2$). The connections are made simultaneously at $t = 0$. Assuming no heat loss to the surroundings, what is the rate of heat flow from the $100^\circ\text{C}$ bath into the first rod at time $t = 80 \ln 2 \text{ s}$?

A spherical shell of inner radius $a$ and outer radius $b$ has a radially varying thermal conductivity given by $k(r) = \frac{k_0 a}{r}$, where $k_0$ is a positive constant. The inner surface at $r=a$ is maintained at temperature $T_1$ and the outer surface at $r=b$ is maintained at temperature $T_2$ ($T_1 > T_2$). In steady state, what is the temperature $T(r)$ at a distance $r$ ($a < r < b$) from the center?

A metallic sphere of mass $m$ is placed in a vacuum chamber maintained at absolute zero. It loses heat to the surroundings via a thermal link such that the rate of heat loss is $P_{loss} = \gamma T$, where $T$ is the absolute temperature of the sphere and $\gamma$ is a positive constant. Simultaneously, the sphere is subjected to a time-varying magnetic field that induces eddy currents. The power dissipated by these eddy currents is $P_{eddy} = \frac{\beta}{T}$, where $\beta$ is a positive constant (this inverse dependence arises because the electrical resistivity of the metal is proportional to its absolute temperature, $\rho \propto T$). The specific heat capacity of the metal varies with temperature as $s(T) = s_0 T$. If the sphere is initially at a temperature $T_0 = 2\sqrt{\frac{\beta}{\gamma}}$, what is the time $t$ required for it to cool down to $T_f = \sqrt{\frac{2\beta}{\gamma}}$?

A thermally insulated vessel contains a mixture of water and $1 \text{ kg}$ of ice at $0^\circ\text{C}$. The total mass of the vessel and its contents is $8 \text{ kg}$. A projectile of mass $2 \text{ kg}$ traveling horizontally at $1000 \text{ m/s}$ strikes the vessel and embeds within it. The vessel is free to move on a frictionless horizontal surface. The collision is perfectly inelastic and generates heat internally within the vessel's thermal control volume. Assume all lost mechanical energy is instantaneously converted to heat. The total heat capacity of the system (vessel, mixture, and embedded projectile) is $C = 10,000 \text{ J/}^\circ\text{C}$ (assumed constant, excluding the latent heat of ice). The latent heat of fusion of ice is $L_f = 3.2 \times 10^5 \text{ J/kg}$. Immediately after the ice melts and thermal equilibrium is reached, the vessel's insulation is removed, exposing it to surroundings at $0^\circ\text{C}$. The system then cools according to Newton's Law of Cooling with a cooling constant $k = 0.1 \text{ min}^{-1}$. What is the temperature of the system $10 \ln 2$ minutes after the insulation is removed?

A thermally isolated radioactive sample of initial mass $m_0$ and constant specific heat capacity $c$ is placed in a vacuum at absolute zero. The sample generates heat at a rate $P_{\text{in}}(t) = P_0 e^{-\lambda t}$ due to radioactive decay, and its mass decreases as $m(t) = m_0 e^{-\lambda t}$. The sample loses energy via Stefan-Boltzmann radiation from its constant surface area $A$, such that $P_{\text{out}}(t) = \sigma A T^4$. The temperature $T$ of the sample is governed by the energy balance $P_{\text{in}}(t) = P_{\text{out}}(t) + m(t) c \frac{dT}{dt}$. If the initial temperature of the sample is $0\text{ K}$, what is the value of $\frac{d^2T}{dt^2}$ at $t=0$?

A solid spherical projectile of radius $R$ and emissivity $e$ moves at a constant high velocity $v$ through an atmosphere of constant density $\rho$. The aerodynamic drag force acting on the projectile is given by $F_d = \frac{1}{2} \rho v^2 C_d A_c$, where $C_d$ is the drag coefficient and $A_c$ is the effective cross-sectional area. Assume that the entire mechanical power dissipated by the drag force is absorbed by the projectile as internal thermal energy. If the projectile loses heat solely via thermal radiation to the surrounding environment (assumed to be at $0\text{ K}$), which of the following expressions represents the steady-state terminal temperature $T$ of the projectile? (Here, $\sigma$ is the Stefan-Boltzmann constant).

A long cylindrical metal rod of initial radius $r_0$ and length $L_0$ is placed between two perfectly rigid, unyielding walls such that its length cannot change. Assume the rod is initially unstressed at $0 \text{ K}$ and its material properties (emissivity $\epsilon$, coefficient of linear thermal expansion $\alpha$, Young's modulus $Y$, and Poisson's ratio $\nu$) are temperature-independent. It is heated by a constant internal power source $P$ and radiates heat to the surrounding vacuum at $0 \text{ K}$. Neglecting heat loss from the flat ends and assuming $\alpha T \ll 1$, what is the steady-state temperature $T$ of the rod to first order in $\alpha$?

(Let $T_0 = \left( \frac{P}{\epsilon \sigma 2\pi r_0 L_0} \right)^{1/4}$, where $\sigma$ is the Stefan-Boltzmann constant).

A spherical blackbody source of proper temperature $T_0$ moves directly towards a stationary observer with a constant velocity $v = \frac{4}{5}c$. In the rest frame of the source, the spectral radiance peaks at a wavelength $\lambda_0$. The observer measures the incoming radiation. Let $\lambda_{obs}$ be the observed peak wavelength. Let $I_{obs}$ be the total observed radiance (energy per unit area per unit time per unit solid angle) integrated over all frequencies, and $I_0$ be the corresponding total proper radiance. Given that the quantity $\frac{I_\nu}{\nu^3}$ is Lorentz invariant (where $I_\nu$ is the spectral radiance and $\nu$ is the frequency), determine the values of $\frac{\lambda_{obs}}{\lambda_0}$ and $\frac{I_{obs}}{I_0}$.

An ideal diatomic gas is confined in a perfectly insulated cylinder with a movable piston. The gas is in thermal equilibrium and its radiation spectrum approximates that of a perfect blackbody. Initially, the internal energy of the gas is $U_0$ and its spectral emissive power peaks at a wavelength $\lambda_0$. The gas undergoes an adiabatic expansion until the peak wavelength of its emitted radiation shifts to $2\lambda_0$. Which of the following represents the work done by the gas and the ratio of its final volume to initial volume $\left(\frac{V_f}{V_0}\right)$, respectively?

A metallic block of heat capacity $C$ oscillates horizontally on a frictionless surface, attached to a spring. It passes through a uniform magnetic field, experiencing a damping force due to eddy currents. As a result, its mechanical energy decays as $E(t) = E_0 e^{-t/\tau_1}$, where $E_0$ is the initial energy. The work done against the damping force is entirely converted into heat within the block. Simultaneously, the block loses heat to the surroundings (at a constant temperature $T_0$) according to Newton's law of cooling, with a thermal time constant $\tau_2$. If $\tau_2 = 2\tau_1$, and the block is initially at temperature $T_0$, what is the maximum temperature reached by the block?

A solid cylindrical rod of length $L$, cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$ is rigidly clamped between two immovable walls at an ambient temperature $T_0$. It is heated via a calorimetry setup involving a hot liquid, reaching an initial uniform temperature $T_0 + \Delta T_0$. The combined heat capacity of the rod (excluding elastic energy) and the liquid is $C_0$. The system is then allowed to cool. The calorimeter loses heat to the surroundings at a rate $\frac{dQ}{dt} = -k(T - T_0)$, where $k$ is a positive constant and $T$ is the instantaneous temperature. Assuming the rod and liquid are always in thermal equilibrium, what is the time $t$ taken for the system to cool to $T_0 + \frac{\Delta T_0}{2}$?

A solid block of volume $V_0$ and density $d$ is completely submerged in a liquid and suspended by a vertical string. The liquid has an initial density $\rho_0$ (where $\rho_0 < d$), mass $m$, specific heat capacity $c$, and coefficient of volume expansion $\beta$. The block's thermal expansion is negligible. At $t=0$, a heater starts supplying heat to the liquid at a constant power $P$. The liquid loses heat to the surroundings at a rate $\lambda(T - T_0)$, where $T$ is the instantaneous temperature of the liquid, $T_0$ is the constant ambient temperature, and $\lambda$ is a positive constant. Assuming uniform temperature in the liquid and $\beta(T-T_0) \ll 1$, which of the following expressions represents the tension in the string as a function of time $t$?

A battery of electromotive force $E = 20 \text{ V}$ and internal resistance $r = 2 \ \Omega$ is connected across a heating coil placed in a calorimeter. The resistance of the coil varies with the temperature elevation $\Delta T$ above the surroundings as $R(\Delta T) = R_0(1 + \alpha \Delta T)$, where $R_0 = 2 \ \Omega$ and $\alpha = 0.05 \text{ K}^{-1}$. The calorimeter loses heat to the surroundings at a rate $P_{\text{loss}} = k \Delta T$, where $k = \frac{15}{16} \text{ W/K}$. Assuming the surroundings are at a constant temperature, what is the steady-state temperature elevation $\Delta T$ of the calorimeter?

A perfectly insulated cylinder is fitted with a movable, frictionless piston. The cylinder contains $n$ moles of an ideal gas and a mass $m$ of ice. Initially, the system is in thermal equilibrium at temperature $T_0$ (the melting point of ice) and the gas pressure is $P_0$. The piston is slowly pushed inward by an external force until all the ice just melts. The gas and the ice-water mixture remain in thermal equilibrium throughout the process. Let $L$ be the specific enthalpy of fusion of ice (latent heat), and $v_i, v_w$ be the specific volumes of ice and water, respectively (assumed constant). Which of the following expressions gives the final pressure of the gas?

A projectile of mass $m$ moving with velocity $v_0$ strikes a block of ice of mass $M$ (initially at rest at $0^\circ\text{C}$) and gets embedded in it. The collision takes place in an environment at constant atmospheric pressure $P_0$. The specific internal energy of fusion of ice (change in internal energy per unit mass during melting) is $u_f$. The densities of ice and water are $\rho_i$ and $\rho_w$ respectively ($\rho_i < \rho_w$). Assuming no heat is exchanged with the surroundings and all the dissipated mechanical energy is retained within the ice-projectile system, what is the mass of ice $\Delta m$ that melts?

A spherical ice nanoparticle of radius $R$ is in phase equilibrium with surrounding liquid water at atmospheric pressure $P_0$. The surface tension of the ice-water interface is $\gamma$. Due to surface reconstruction, the density of ice at the boundary layer (where melting occurs) is $\rho_s$, which differs from the bulk ice density $\rho_i$. The density of bulk water is $\rho_w$. The latent heat of fusion per unit mass is $L$, and the normal melting point at $P_0$ is $T_0$. Assuming the temperature shift $\Delta T \ll T_0$, which of the following expressions correctly represents the shift in the melting point $\Delta T$ of the nanoparticle?

A mass $m$ of a liquid boils and transforms into vapor at a constant temperature $T$ and pressure $P$. The latent heat of vaporization of the liquid is $L$. If the slope of the liquid-vapor phase boundary in the pressure-temperature ($P-T$) diagram at this state is given by $\frac{dP}{dT}$, which of the following expressions correctly represents the change in internal energy ($\Delta U$) of the system during this complete phase transition?

A closed cubical container of side length $H$ is completely filled with a liquid of total mass $M$ and density $\rho$. The container is initially at rest, and the liquid is at its normal boiling point $T_0$ with the pressure at the top-front edge ($x=0, y=0$) being $P_0$. The container is then given a constant horizontal acceleration $a$ in the $-x$ direction (where $y$ is the downward vertical direction). The rate of change of the boiling point with pressure is a constant $\alpha$ (i.e., $\frac{dT}{dP} = \alpha$). To maintain the liquid at the verge of boiling everywhere and to vaporize a small uniform mass fraction $f$ of the liquid throughout the volume, a total heat $Q$ must be supplied. Assume the pressure at the top-front edge remains $P_0$, and the volume of vapor formed is negligible. Given $C$ is the specific heat capacity of the liquid and $L_v$ is the latent heat of vaporization, find the total heat $Q$ supplied.

Three identical cylindrical rods, each of length $L$ and cross-sectional area $A$, are joined at a common junction $J$ to form a star network. The free ends of the rods are maintained at constant temperatures $T_1 = 500 \text{ K}$, $T_2 = 100 \text{ K}$, and $T_3 = 100 \text{ K}$. The thermal conductivity of the material of the rods varies with absolute temperature $T$ as $k = \alpha T$, where $\alpha$ is a positive constant. Assuming steady-state heat flow and no heat loss from the lateral surfaces of the rods, what is the temperature of the junction $J$?

A uniform rod of length $L$ and cross-sectional area $A$ is rigidly clamped between two fixed walls at $x=0$ and $x=L$. The Young's modulus of the rod is $Y$. The thermal conductivity of the rod varies with position as $K(x) = K_0 \left(1 + \frac{x}{L}\right)$, and the coefficient of linear thermal expansion varies as $\alpha(x) = \frac{\alpha_0}{1 + \frac{x}{L}}$. Initially, the entire rod is at a uniform temperature $T_i$. The end at $x=0$ is maintained at $T_i$, and the end at $x=L$ is maintained at $T_i + T_0$. Assuming steady-state heat flow is established and the walls do not yield, what is the thermal stress $\sigma$ developed in the rod?

Consider a blackbody cavity of volume $V$ at an equilibrium temperature $T$. Due to photon neutrality, the number of photons is not conserved, and their chemical potential is zero. Let $E_w$ be the energy of a single photon corresponding to the wavelength $\lambda_{max}$ at which the spectral energy density per unit wavelength is maximum. Let $\langle E \rangle$ be the average energy of a photon in the cavity. If $x$ is the non-zero root of the equation $x = 5(1 - e^{-x})$, which of the following expressions correctly represents the ratio $\frac{E_w}{\langle E \rangle}$? \ (Given: $\int_0^\infty \frac{y^{s-1}}{e^y - 1} dy = \Gamma(s)\zeta(s)$, where $\zeta(s)$ is the Riemann zeta function, $\Gamma(4)=6$, $\Gamma(3)=2$, and $\zeta(4)=\frac{\pi^4}{90}$)

A cylindrical rod of length $L$ and cross-sectional area $A$ is placed between two heat reservoirs maintained at temperatures $T_1 = 800 \text{ K}$ (at $x = 0$) and $T_2 = 200 \text{ K}$ (at $x = L$). The thermal conductivity of the rod varies with position $x$ as $k(x) = k_0 \left(1 + \frac{x}{L}\right)^2$, where $k_0$ is a positive constant. The lateral surface of the rod is perfectly insulated. In the steady state, a small hole is made at the midpoint of the rod ($x = L/2$) to observe the emitted blackbody radiation. If Wien's displacement constant is $b$, what is the wavelength $\lambda_{\text{max}}$ corresponding to the maximum spectral emissive power of the radiation emitted from the midpoint?