Mechanical Properties of Solids - Class 11 Part 2

One mole of an ideal monoatomic gas is used to model the energy dissipation of an elastic material undergoing a hysteresis cycle. The gas undergoes a three-step cyclic process $A \to B \to C \to A$:

1. Expansion (stretching analog) from state $A(P_0, V_0)$ to $B(2P_0, 2V_0)$ following a polytropic process $P \propto V$.
2. Compression (relaxation analog) from state $B$ to $C(P_C, V_0)$ following a polytropic process $P \propto V^2$.
3. Isochoric return from state $C$ to state $A$. Let $Q_{\text{rej}}$ be the magnitude of the heat rejected by the gas during the relaxation phase ($B \to C$), and $W_{\text{net}}$ be the net work done by the gas in the entire cycle (representing the hysteresis loss). What is the ratio $\frac{Q_{\text{rej}}}{W_{\text{net}}}$?

A particle of mass $m$ is constrained to move along the x-axis. It is subjected to a force that consists of a static destabilizing component and a viscoelastic stabilizing memory component, given by: $F(t) = k x(t) - \int_0^t \beta e^{-\gamma(t-\tau)} x(\tau) d\tau$ where $k$, $\beta$, and $\gamma$ are strictly positive constants, and $x(t)$ is the displacement from the equilibrium position $x=0$. The particle is released from rest at $t=0$ with a small displacement $x_0$. Which of the following conditions on $\beta$ ensures that the equilibrium position $x=0$ is dynamically stable?

A cylindrical metal rod of Young's modulus $E$, Poisson's ratio $\nu$, and coefficient of linear thermal expansion $\alpha$ is constrained longitudinally between two rigid, immovable walls. Laterally, it is enclosed by a non-rigid sleeve that exerts a radial pressure $p = \beta E \epsilon_r$, where $\epsilon_r$ is the lateral strain and $\beta$ is a positive dimensionless constant. If the rod is heated by a temperature $\Delta T$, what is the fractional change in its density, $\frac{\Delta \rho}{\rho_0}$? (Assume small strains and $\rho_0$ is the initial density).

A ferromagnetic isotropic material with Young's modulus $Y$ and Poisson's ratio $\sigma$ is placed in a uniform external magnetic field, reaching a saturation magnetization $M_s$. The magneto-elastic coupling contributes an energy density $U_{me} = -\frac{1}{2} \mu_0 \gamma M_s^2 \theta^2$, where $\theta$ is the volumetric strain and $\gamma$ is a positive dimensionless constant. If the material is subjected to a small external hydrostatic pressure $P$, what is the magnitude of the equilibrium volumetric strain $|\theta|$, and what condition must $\sigma$ satisfy for the material to remain mechanically stable under this magnetic saturation? (Assume the intrinsic thermodynamic limit $\sigma < 1/2$ holds).

A cylindrical rod of a novel metamaterial has a constant bulk modulus $K = 1.0 \times 10^9 \text{ N/m}^2$. During a slow 1D longitudinal tensile test, its Poisson's ratio $\nu$ varies with the longitudinal strain $\epsilon$ as $\nu(\epsilon) = 0.25 - 10\epsilon$. The longitudinal stress $\sigma$ and strain $\epsilon$ are related by $\sigma = Y(\epsilon)\epsilon$, where $Y(\epsilon) = 3K(1 - 2\nu(\epsilon))$ is the effective Young's modulus. If the rod is stretched from an unstrained state ($\epsilon = 0$) to a final strain of $\epsilon = 0.05$, what is the work done per unit volume (strain energy density) on the rod?

Two long, thin cylindrical rods, Rod 1 and Rod 2, are joined end-to-end at $x=0$. Both rods have the same uniform cross-sectional area and the same uniform density $\rho$. The materials of both rods have an identical shear modulus $G$. Rod 1 has a bulk modulus $B_1 = \frac{2}{3}G$. Rod 2 is made of an ideal incompressible material ($B_2 \to \infty$). A longitudinal displacement wave $s_i(x,t) = A_i \sin(\omega t - k_1 x)$ travels through Rod 1 towards the junction. If the reflected wave is $s_r(x,t) = A_r \sin(\omega t + k_1 x)$, what is the amplitude reflection coefficient $R = \frac{A_r}{A_i}$?

A uniform metal bar of length $L$, cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear expansion $\alpha$ is placed on a smooth horizontal surface. One end of the bar is fixed to a rigid wall, and the other end is attached to a massless ideal spring of force constant $k$. The other end of the spring is fixed to another rigid wall. Initially, the spring is in its relaxed state and the bar is at a uniform temperature $T_0$. The temperature of the bar is then slowly increased by $\Delta T$. Assuming $Y$ and $\alpha$ are independent of temperature, what is the elastic potential energy stored in the metal bar due to thermal stress?

A uniform thin circular ring of mass $m$ and radius $R$ is rigidly attached to a massless, non-expanding rod of length $R$. The rod is collinear with a diameter of the ring, connecting the ring's circumference to a frictionless pivot $P$. The system oscillates as a physical pendulum in a vertical plane with a small time period $T_0$. The ring is made of a material with a coefficient of linear thermal expansion $\alpha$. If the temperature of the ring is increased by $\Delta T$ (while the rod's length remains unaffected), the new time period becomes $T$. Assuming $\alpha \Delta T \ll 1$, the fractional change in the time period $\frac{T - T_0}{T_0}$ is:

A small block of mass $m = 1 \text{ kg}$ is attached to one end of a light elastic string of natural length $L_0 = 1 \text{ m}$ and stiffness $k = 12 \text{ N/m}$. The other end of the string is fixed to a point on a smooth horizontal table. The block is initially revolving in a horizontal circle with an angular velocity $\omega_1 = 2 \text{ rad/s}$. An external mechanism slowly increases the angular velocity of the block to $\omega_2 = 3 \text{ rad/s}$. Calculate the work done by the external mechanism during this process.

An ideal elastic string of natural length $L$ and stiffness $k$ lies along the $x$-axis with one end fixed at the origin. A small electric dipole of moment $p$ is attached to the free end of the string. The region contains a non-uniform electric field $\vec{E} = \alpha x^2 \hat{i}$ (where $\alpha > 0$). Initially, the dipole is held at $x = L$ with its moment oriented along the $y$-axis ($\vec{p} = p \hat{j}$). An external agent slowly rotates the dipole until it is aligned with the positive $x$-axis ($\vec{p} = p \hat{i}$) and simultaneously allows the string to reach its new equilibrium length. Assuming $k > 2p\alpha$ and all quantities are in SI units, what is the work done by the external agent during this entire process?

A solid block of volume $V$, height $H$, and density $\rho_b$ is suspended in air by a light elastic string of stiffness $k$ attached to a rigid support. A wide container is placed below the block and is slowly filled with a liquid of density $\rho$ (where $\rho < \rho_b$). As the liquid level rises, the block moves upwards due to the contraction of the string. The liquid is poured until the block is just completely submerged. What is the total work done by the buoyant force on the block during this process?

A viscoelastic material is subjected to a cyclic stress $\sigma(t) = \sigma_0 \sin(\omega t)$. Due to the elastic after-effect, the material does not respond instantaneously. The strain $\epsilon(t)$ and strain rate $\dot{\epsilon}(t)$ are related to the applied stress by the Kelvin-Voigt model equation: $\eta \dot{\epsilon} + k \epsilon = \sigma(t)$, where $\eta$ is the coefficient of internal viscosity and $k$ is the elastic modulus. After the system reaches a steady state, what is the total energy dissipated per unit volume of the material during one complete cycle of loading?

A viscoelastic wire of volume $V$ (in $m^3$) is subjected to a cyclic tensile stress $\sigma(t) = \sigma_0 \sin(\omega t)$ (in $N/m^2$). Due to the elastic after-effect, the strain lags the stress and is given by $\varepsilon(t) = \varepsilon_0 \sin(\omega t - \delta)$, where $\delta$ is the phase angle. Assuming the process is isothermal and all dissipated mechanical energy appears as heat, what is the total heat generated in the wire during one complete cycle?

An electromechanical system consists of a solenoid with a non-conducting, compressible magnetic core, connected in series with a resistor of resistance $R$ and a DC voltage source $V$. For $t < 0$, a constant mechanical compressive force is applied to the core, and the circuit is in a steady state. At $t = 0$, the compressive force is suddenly removed. Due to the elastic after-effect, the core's strain recovers gradually, causing the inductance of the solenoid to vary with time as $L(t) = L_1 + L_2 (1 - e^{-t/\tau})$, where $L_1, L_2$, and $\tau$ are positive constants. Which of the following represents the rate of change of the magnetic energy stored in the solenoid exactly at $t = 0^+$?

A viscoelastic fiber exhibits an elastic after-effect modeled by the Kelvin-Voigt equation: $\sigma = Y \epsilon + \eta \frac{d\epsilon}{dt}$, where $Y$ is the Young's modulus and $\eta$ is the viscosity. The relaxation time $\tau = \eta/Y$ depends on the absolute temperature $T$ according to $\tau(T) = \tau_0 e^{E_a / k_B T}$, where $E_a$ is the activation energy and $k_B$ is the Boltzmann constant. Initially, a constant stress $\sigma_0$ is applied for a long time at temperature $T_0$, producing a steady strain $\epsilon_0$. At $t=0$, the stress is suddenly removed ($\sigma = 0$), and simultaneously, the fiber is heated such that its temperature increases linearly with time: $T(t) = T_0 + \alpha t$, where $\alpha > 0$ is a small constant. Assuming $\alpha t \ll T_0$ and $\frac{E_a \alpha t}{k_B T_0^2} \ll 1$, find the expression for the strain $\epsilon(t)$ during the early stages of recovery. Let $\tau_1 = \tau(T_0)$.

A thick wire of length $L$, cross-sectional area $A$, and Young's modulus $Y$ is suspended vertically from a rigid support. The linear mass density of the wire is non-uniform and varies as $\lambda(x) = kx^2$, where $x$ is the distance measured from its free lower end and $k$ is a positive constant. If the total mass of the wire is $M$ and the acceleration due to gravity is $g$, what is the total elongation of the wire under its own weight?

A uniform heavy wire of mass $M$, length $L$, cross-sectional area $A$, and Young's modulus $Y$ is suspended vertically from a rigid support. Acceleration due to gravity is $g$. What is the total elastic potential energy stored in the wire due to its own weight?

A uniform heavy elastic wire of mass $M$, natural length $L$, cross-sectional area $A$, and Young's modulus $Y$ is suspended vertically from a rigid ceiling. A block of mass $m$ is attached to its lower end. The block is pulled down slightly and released, causing longitudinal oscillations. Assuming the dynamic displacement profile along the wire is linear, what is the angular frequency $\omega$ of the oscillations?

A uniform heavy wire of length $L$, cross-sectional area $A$, density $\rho_w$, and Young's modulus $Y$ hangs vertically from a rigid support. The wire is completely submerged in a non-homogeneous liquid whose density varies with depth $y$ (measured downwards from the support) as $\rho(y) = \rho_0(1 + \alpha y)$, where $\rho_0$ and $\alpha$ are positive constants. Assuming the wire remains taut, what is the total steady-state elongation of the wire?

A uniform rod of length $L$ and cross-sectional area $A$ is rigidly fixed between two immovable walls at $x=0$ and $x=L$. Initially, the rod is unstressed at a uniform temperature $T_0$. The temperature of the rod is then changed to a steady state distribution $T(x) = T_0(1 + k x^2)$, where $k$ is a positive constant. The coefficient of linear thermal expansion of the rod is $\alpha$. The Young's modulus of the rod varies with position as $Y(x) = \frac{Y_0}{1 + \beta x^2}$, where $Y_0$ and $\beta$ are positive constants. What is the compressive force $F$ developed in the rod?

A small block of mass $m$ is attached to one end of a uniform wire of natural length $L_0$, cross-sectional area $A$, and Young's modulus $Y$. The other end of the wire is fixed to a vertical axis. The block rests on a smooth horizontal surface and is constrained to move along a radial groove in a horizontal disc. The disc is rotated by a motor at a constant angular velocity $\omega$. Initially, the block is held at a distance $L_0$ from the axis and then released. Due to internal friction in the wire, the block eventually comes to rest relative to the disc. Assuming $YA > m \omega^2 L_0$, what is the total energy dissipated as heat due to internal friction?

A cylindrical metallic wire of length $L_0$, radius $r$, and emissivity $\epsilon$ hangs vertically from a rigid support, carrying a mass $M$ at its lower end. At ambient temperature $T_0$, the free electron density is $n$ and the relaxation time is $\tau_0$. When a constant current $I$ flows through the wire, its temperature increases by $\Delta T$. The relaxation time decreases as $\tau = \tau_0 / (1 + \beta \Delta T)$, and the Young's modulus decreases as $Y = Y_0 - \alpha \Delta T$. The wire loses heat only via radiation to the surroundings at $T_0$ (Stefan-Boltzmann constant $\sigma_s$). Assuming $\Delta T \ll T_0$ and ignoring linear thermal expansion, what is the steady-state increase in the length of the wire due to the current? (Let $m_e$ and $e$ be the mass and charge of an electron).

A cubic semiconductor crystal is subjected to a uniform tensile stress $\sigma$ along the $[100]$ crystallographic axis. The elastic stiffness constants of the crystal are given by $C_{11}$ and $C_{12}$ in Voigt notation. According to the deformation potential theory, the shift in the energy band gap is given by $\Delta E_g = a \cdot \text{Tr}(\epsilon)$, where $a$ is the hydrostatic deformation potential and $\epsilon$ is the strain tensor. Which of the following expressions correctly represents the shift in the energy band gap $\Delta E_g$?

A mechanical spring undergoing cyclic deformation exhibits elastic hysteresis, dissipating energy $\Delta E = \alpha x_0^2$ per cycle, where $x_0$ is the displacement amplitude. In a direct electromechanical analogy, this spring is modeled as an inductor with a lossy material core in a series $RLC$ circuit. The displacement $x$ maps to the charge $q$ on the capacitor, and the velocity $v$ maps to the current $i$. The hysteresis loss is represented by an equivalent series resistance $R$. If $\alpha_e$ is the electrical analog of the constant $\alpha$, what is the value of $R$ in terms of $\alpha_e$, the inductance $L$, and the capacitance $C$? Assume the damping is small.

A hypothetical crystalline solid has a simple cubic lattice with a lattice constant $a$ at $0 \text{ K}$. The interatomic potential energy $U(x)$ between adjacent atoms is modeled as: $U(x) = \frac{1}{2} c x^2 - \frac{1}{3} g x^3$ where $x = r - a$ is the displacement from the $0 \text{ K}$ equilibrium separation, and $c$ and $g$ are positive constants. The solid is heated to a temperature $T$ (such that $k_B T \ll c a^2$, where $k_B$ is the Boltzmann constant) and allowed to expand freely. The effective Young's modulus of the solid at temperature $T$ is defined as $Y(T) = \frac{1}{a} \left. \frac{d^2 U}{dx^2} \right|_{x = \langle x \rangle}$, where $\langle x \rangle$ is the mean interatomic displacement at temperature $T$. Assuming classical statistical mechanics applies, which of the following expressions correctly represents $Y(T)$? (Given: $\int_{-\infty}^{\infty} x^4 e^{-\alpha x^2} dx = \frac{3}{4\alpha^2} \sqrt{\frac{\pi}{\alpha}}$ and $e^z \approx 1 + z$ for small $z$)

Two identical spherical molecules, each of mass $m$ and physical radius $R$, are initially at rest at an infinite separation. They attract each other via a dipole-dipole interaction, but at short distances, the Pauli exclusion principle causes a strong core repulsion. The net effective potential energy of the two-molecule system is given by $U(r) = \frac{A}{r^6} - \frac{B}{r^3}$, where $r$ is the center-to-center distance, and $A$ and $B$ are positive constants. Given that $\left(\frac{A}{B}\right)^{1/3} > 2R$, what is the maximum relative momentum $p_{\text{rel}}$ achieved by the molecules, and what is their distance of closest approach $r_{\text{min}}$?

A block of mass $M$ slides at a constant low velocity $v$ over a rough horizontal surface. The macroscopic normal force is $N = Mg$. Due to microscopic surface roughness, the true area of contact is $A_t = N / P_Y$, where $P_Y$ is the yield pressure of the softer material. The interface forms adhesive bonds with an area density of $\sigma$ bonds per unit true contact area. Each bond behaves as an ideal linear spring with force constant $k$ and breaks when stretched by a critical distance $x_c$. After breaking, the bond dissipates its stored elastic energy entirely as heat and immediately reforms at the next lattice site, located at a distance $a$ along the sliding direction. Which of the following expressions correctly represents the coefficient of kinetic friction $\mu_k$?

A cylindrical rod of length $L$ and constant cross-sectional area $A$ is rigidly clamped between two immovable walls at $x=0$ and $x=L$. Initially, the rod is at a uniform temperature. The temperature of the rod is then increased such that the temperature change at any cross-section $x$ is given by $\Delta T(x) = T_0 \frac{x}{L}$, where $T_0$ is a positive constant. The coefficient of linear thermal expansion of the material varies with position as $\alpha(x) = \alpha_0 \frac{x}{L}$. Furthermore, the Young's modulus of the material depends on the local temperature change as $Y(\Delta T) = \frac{Y_0}{1 + \frac{\Delta T}{T_0}}$. Assuming the rod does not buckle, what is the magnitude of the compressive force $F$ developed in the rod?

A block of mass $m = 2 \text{ kg}$ is suspended vertically from a specialized non-linear elastic rod. The restoring force exerted by the rod when stretched by a distance $x$ from its natural length is given by $F(x) = -(20x + 80x^3) \text{ N}$, where $x$ is in meters. The block is allowed to reach its equilibrium position. It is then given an infinitesimally small vertical displacement and released. Assuming the acceleration due to gravity is $g = 10 \text{ m/s}^2$, what is the time period of the resulting small oscillations?

A uniform elastic rod of length $L$, cross-sectional area $A$, and Young's modulus $Y$ is initially unstretched. One end is fixed to a rigid wall. At $t=0$, a motorized mechanism attached to the free end begins to stretch the rod. The mechanism is programmed to deliver a transient mechanical power $P(t) = P_0 e^{-t/\tau}$ entirely into the rod, where $P_0$ and $\tau$ are positive constants. Assume the rod's mass is negligible, the process is isothermal, and all delivered work is stored as elastic potential energy. If the breaking stress of the rod's material is $\sigma_b$, at what time $t_b$ will the rod break? (Assume $2 Y P_0 \tau > A L \sigma_b^2$).

A vertical rod of length $L$ and constant cross-sectional area $A$ hangs freely from a rigid support at its upper end. The material of the rod has a constant Young's modulus $Y$. The density of the rod varies with the distance $y$ from its lower free end according to the relation $\rho(y) = \rho_0 \left(1 + \frac{y}{L}\right)^2$, where $\rho_0$ is a positive constant. Let $\Delta L$ be the total elongation of the rod under its own weight in equilibrium. A longitudinal elastic wave is generated at the lower end and takes time $T$ to travel to the upper support. Find the ratio $\frac{\Delta L}{T^2}$. (Assume that the stress is within the proportional limit and $g$ is the acceleration due to gravity).

A straight rod of length $L$ is rigidly fixed between two unyielding walls. The rod has a uniform Young's modulus $E$. Its cross-sectional area $A(x)$ and coefficient of linear thermal expansion $\alpha(x)$ vary along its length $x$ (from $x=0$ to $x=L$) as: $A(x) = A_0 2^{-x/L}$ $\alpha(x) = \alpha_0 2^{x/L}$ where $A_0$ and $\alpha_0$ are positive constants. The temperature of the rod is uniformly increased by $\Delta T$. If the material fails when the local compressive stress reaches a critical value $\sigma_b$, at what minimum temperature rise $\Delta T$ will the rod break?

A block of mass $m$ is suspended in equilibrium from a rigid support by a vertical, uniform, massless wire of length $L$, cross-sectional area $A$, and Young's modulus $Y$. The breaking stress of the wire is $\sigma_0$ (where $\sigma_0 > \frac{mg}{A}$). A sudden downward impulse $J$ is imparted to the block. Assuming the wire behaves as a linear spring and does not slacken, what is the maximum value of $J$ such that the wire does not break during the subsequent oscillation?

The interatomic potential energy $U(r)$ (in $\text{J}$) of a hypothetical simple cubic crystal is modeled by the Morse potential: $U(r) = U_0 \left[ \left( 1 - e^{-a(r-r_0)} \right)^2 - 1 \right]$ where $U_0$ is the dissociation energy (in $\text{J}$), $r_0$ is the equilibrium interatomic distance (in $\text{m}$), $a$ is a positive constant (in $\text{m}^{-1}$), and $r$ is the distance between adjacent atoms (in $\text{m}$). Assuming the crystal fractures when the applied tensile force per atomic chain exceeds the maximum possible interatomic attractive force, what is the theoretical macroscopic breaking stress $\sigma_b$ (in $\text{N/m}^2$) of the material? (Assume the cross-sectional area per atomic chain remains $r_0^2$ up to the fracture point).

A thin, flexible insulating ring of radius $R$ and cross-sectional area $A$ has a uniform linear mass density $\lambda$ and a uniform positive linear charge density $\mu$. The ring is rotated about its central axis, which is perpendicular to its plane, with a constant angular velocity $\omega$. A uniform magnetic field of magnitude $B$ is applied parallel to the axis of rotation such that the resulting Lorentz force acts radially outward. If the material of the ring has a breaking stress $\sigma_b$, which of the following expressions gives the minimum magnetic field $B$ required to break the ring? (Assume the ring remains perfectly circular until it breaks and neglect gravity).

A cylindrical metallic wire of length $L$ and cross-sectional area $A$ is subjected to a cyclic tensile stress. Due to the elastic after-effect, the longitudinal strain $\epsilon$ lags behind the applied stress $\sigma$, forming a hysteresis loop in the $\sigma-\epsilon$ plane. The area of this hysteresis loop is $S$ (in $\text{J m}^{-3}$). The wire is maintained at a constant temperature $T$ by a surrounding heat bath. Which of the following statements regarding one complete cycle is INCORRECT?

Two identical large, thin, square elastic insulating sheets, each of initial side length $L$, are placed parallel to each other at a small separation $d$ ($d \ll L$). They carry uniformly distributed total charges $+Q$ and $-Q$ respectively. The sheets have Young's modulus $Y$, Poisson's ratio $\nu$, and coefficient of linear thermal expansion $\alpha$. Each sheet is confined between two smooth, immovable rigid walls separated by distance $L$, which completely prevent any expansion in the x-direction but allow free frictionless sliding in the y-direction. Initially, the system is at temperature $T_0$ and the sheets are unstressed. If the temperature of both sheets is increased by $\Delta T$, what is the new electrostatic force of attraction $F$ between the sheets? (Assume $\alpha \Delta T \ll 1$ and neglect any deformation caused by the electrostatic force itself).

A metallic rod of initial length $L_0$ (in $\text{m}$) and cross-sectional area $A$ (in $\text{m}^2$) is at an initial temperature $T_0$ (in $\text{K}$). Its coefficient of linear thermal expansion is $\alpha$ (in $\text{K}^{-1}$). The Young's modulus of the material varies with temperature $T$ as $Y(T) = \frac{Y_0}{1 + \beta (T - T_0)}$, where $Y_0$ (in $\text{N m}^{-2}$) and $\beta$ (in $\text{K}^{-1}$) are positive constants. A tensile force $F$ (in $\text{N}$) is gradually applied to the rod, increasing from $0$ to $F_0$. Simultaneously, the temperature of the rod is increased from $T_0$ to $T_0 + \Delta T$. The force is increased such that it is always proportional to the temperature change, i.e., $F(T) = \frac{F_0}{\Delta T} (T - T_0)$. Assuming the rod is free to expand and strains are small, what is the total work done (in $\text{J}$) by the applied force $F$ during this process?

A metallic rod of initial length $L_0$ and cross-sectional area $A$ is initially at temperature $T_0$. Its Young's modulus varies with temperature as $Y(T) = \frac{Y_0}{1 + \beta (T - T_0)}$, where $Y_0$ and $\beta$ are positive constants. The coefficient of linear thermal expansion of the rod is $\alpha$. The rod is subjected to a thermodynamic process where a tensile force $F$ is gradually increased from $0$ to $F_0$ while the temperature is simultaneously increased from $T_0$ to $T_0 + \Delta T$. The force and temperature are coupled such that $F(T) = F_0 \frac{T - T_0}{\Delta T}$. Assuming the strains remain small and the rod is always in mechanical equilibrium, what is the total work done by the external force on the rod during this process?

Two long, thin cylindrical rods, A and B, are joined end-to-end at $x=0$. Both rods have the same uniform cross-sectional area $S$, identical density $\rho$, and identical Bulk Modulus $B$. Rod A has a Poisson's ratio $\sigma_A = 0.25$. Rod B is made of a nearly incompressible material such that its Poisson's ratio $\sigma_B \to 0.5$. A longitudinal harmonic displacement wave travels in Rod A towards the junction. Four wave properties are evaluated in the order given below: (I) Acoustic impedance of Rod A (II) Phase difference between reflected and incident displacement waves at $x=0$ (III) Speed of the transmitted longitudinal wave in Rod B (IV) Ratio of reflected to incident average power

One of the theoretically calculated values for these properties is INCORRECT. Identify the INCORRECT value from the options below (the options correspond to the properties in the same order).

A uniform non-magnetic metal rod of length $L$, cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear expansion $\alpha$ is placed along the x-axis. Its left end is rigidly fixed at $x=0$. A small magnetic dipole of magnetic moment $\vec{\mu} = -\mu \hat{i}$ (where $\mu > 0$) is rigidly attached to the free right end of the rod at $x=L$. The system is placed in a non-uniform magnetic field given by $\vec{B} = B_0 \left[ 1 + \frac{\lambda}{2} (x-L)^2 \right] \hat{i}$ for $x \ge L$, where $B_0$ and $\lambda$ are positive constants. The rod is heated uniformly such that its temperature increases by $\Delta T$. Assuming the magnetic dipole remains aligned with the x-axis and the rod does not buckle, what is the total potential energy stored in the system (the sum of the elastic strain energy in the rod and the work done against the magnetic force)?

A uniform thin circular ring of mass $m$, initial radius $R_0$, uniform cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear expansion $\alpha$ is suspended from a fixed frictionless pivot $P$ located exactly on its circumference. The ring is initially at temperature $T_0$ and performs small oscillations in its own vertical plane. The temperature of the ring is then uniformly increased by $\Delta T$ (where $\alpha \Delta T \ll 1$). Let $f_I = \frac{\Delta I}{I_0}$ be the fractional change in the moment of inertia of the ring about the pivot $P$, $f_\tau = \frac{\Delta \tau}{\tau_0}$ be the fractional change in its time period of small oscillations, and $\sigma$ be the thermal stress developed in the ring due to the temperature change. Which of the following represents the correct tuple $(f_I, f_\tau, \sigma)$?

A cubic semiconductor crystal has independent elastic compliance constants $S_{11}$, $S_{12}$, and $S_{44}$ (in $\text{Pa}^{-1}$). A uniform uniaxial compressive force $F$ is applied to the crystal along the crystallographic direction $[111]$. The cross-sectional area of the crystal perpendicular to the force is $A$.

According to Deformation Potential Theory, the shift in the energy band gap $\Delta E_g$ due to lattice strain is given by $\Delta E_g = a \cdot \text{Tr}(\epsilon)$, where $a$ is the hydrostatic deformation potential (in eV) and $\epsilon$ is the strain tensor.

Assuming the crystal remains in the linear elastic regime and the effective masses of electrons and holes are unaffected by the deformation, what is the isothermal shift in the intrinsic Fermi level relative to the valence band maximum, $\Delta E_F$?

Note: The intrinsic Fermi level is given by $E_F = E_v + \frac{E_g}{2} + \frac{3}{4} k_B T \ln\left(\frac{m_h^*}{m_e^*}\right)$.

A mechanical oscillator consists of a block of mass $m = 1.0 \text{ kg}$ attached to a viscoelastic spring of stiffness $k = 100 \text{ N/m}$. Due to elastic hysteresis, the spring dissipates energy. For an oscillation cycle with amplitude $x_0$, the area of the hysteresis loop in the force-extension graph is given by $A = \pi x_0^2 \text{ J}$. This mechanical system is modeled as an equivalent series $LCR$ circuit, where the mass corresponds to the inductance $L$, the spring stiffness corresponds to the inverse of capacitance $1/C$, and the hysteresis loss is modeled by an equivalent constant resistance $R$. Assuming the damping is small, which of the following statements regarding the system is INCORRECT?

A mechanical damper exhibiting elastic hysteresis is modeled as an ideal gas (with dimensionless adiabatic index $\gamma$) undergoing a thermodynamic cycle. The cycle consists of three reversible processes:

1. Polytropic expansion from state A with pressure $P_0$ (in $\text{Pa}$) and volume $V_0$ (in $\text{m}^3$) to state B at volume $\eta V_0$ with dimensionless polytropic index $n_1$.
2. Isochoric pressure increase from state B to state C.
3. Polytropic compression from state C back to state A with dimensionless polytropic index $n_2$. It is given that $1 < n_2 < n_1 < \gamma$ and $\eta > 1$ (where $\eta$ is a dimensionless constant). During the cycle, net mechanical work is converted into heat. What is the total heat rejected to the surroundings per cycle?

A long isotropic ferromagnetic rod of intrinsic Young's modulus $Y$ and Poisson's ratio $\nu$ is placed in a uniform external magnetic field $H_0$ parallel to its axis. Due to the magnetostriction effect, the magnetic susceptibility $\chi$ of the material depends on the volumetric strain $\theta = \frac{\Delta V}{V_0}$ as $\chi(\theta) = \chi_0 (1 + \lambda \theta^2)$, where $\chi_0$ and $\lambda$ are positive dimensionless constants. The system is maintained at a constant temperature. Which of the following expressions correctly represents the effective Bulk Modulus $K_{eff}$ of the rod for small volumetric strains, and the critical external magnetic field $H_c$ at which the rod becomes mechanically unstable?

A uniform wire of initial length $L_0$, cross-sectional area $A$, and perimeter $p$ is suspended in a vacuum chamber maintained at a constant ambient temperature $T_0$. A constant mechanical tension $F$ is applied to the wire. Simultaneously, a constant direct current $I$ is passed through it. The resistivity of the wire varies with temperature as $\rho(T) = \rho_0 [1 + \beta(T - T_0)]$, where $\rho_0$ is the resistivity at $T_0$. Due to lattice anharmonicity, the Young's modulus of the wire decreases with temperature as $Y(T) = Y_0 [1 - \alpha(T - T_0)]$. The wire loses heat to the surroundings purely by radiation. The emissivity of the wire surface is $\epsilon$, and the Stefan-Boltzmann constant is $\sigma$. Assuming the steady-state temperature rise $\Delta T = T - T_0$ is small ($\Delta T \ll T_0$), what is the steady-state elastic elongation $\Delta L$ of the wire? (Ignore the thermal expansion of the wire and assume the cross-sectional area remains approximately constant).

A uniform heavy wire of mass $M$, length $L$, cross-sectional area $A$, and Young's modulus $Y$ hangs vertically from a rigid support under its own weight. Let $U_{\text{true}}$ be the correct total elastic potential energy stored in the wire. A student INCORRECTLY calculates the stored energy by assuming the elastic energy density is uniform throughout the wire and equal to the local energy density at the wire's mid-point. If this incorrectly calculated total energy is $U_{\text{wrong}}$, what is the value of the difference $(U_{\text{true}} - U_{\text{wrong}})$?

A heavy, thick wire of total mass $M$, length $L$, and uniform cross-sectional area $A$ is suspended vertically from a rigid support. The material of the wire has a Young's modulus $Y$. The linear mass density $\lambda$ of the wire is not uniform; it varies with the downward vertical distance $x$ from the top support according to the relation $\lambda(x) = kx^2$, where $k$ is a positive constant. If $g$ is the acceleration due to gravity, what is the total elongation of the wire due to its own weight?

A cylindrical rod of length $L$ and cross-sectional area $A$ is oriented along the body diagonal of a cubical room of side $a$ (so $L = \sqrt{3}a$). The rod is fixed between two rigid walls at its ends $x=0$ and $x=L$. The rod has a uniform volume charge density $\rho_e$ and is placed in a uniform magnetic field $\vec{B} = B_0 \hat{k}$. The rod is initially at a uniform temperature $T_0$. The temperature is then increased such that the change in temperature at a distance $x$ from the $x=0$ end is $\Delta T(x) = \Delta T_0 \frac{x}{L}$. The coefficient of linear thermal expansion varies as $\alpha(x) = \alpha_0 \frac{x}{L}$. The Young's modulus of the rod varies with temperature as $Y(T) = \frac{Y_0}{1 + \beta(T - T_0)}$. Which of the following statements regarding the rod in its final equilibrium state is INCORRECT?