Oscillations - Class 11 Part 2

A spherical particle of mass $m = 2 \times 10^{-3} \text{ kg}$ and radius $r = 1 \text{ mm}$ is placed in a conservative potential well $U(x) = \alpha x^2$, where $\alpha = 0.025 \text{ J/m}^2$. The well is filled with a viscous liquid of viscosity $\eta = \frac{2}{\pi} \text{ Pa}\cdot\text{s}$. At $t=0$, the particle is displaced to $x = x_0$ and released from rest. The ensuing displacement of the particle is modeled as $x(t) = A e^{-\gamma t} \cos(\omega_d t + \phi)$, where $\omega_d$ is the damped angular frequency and $\phi \in (-\frac{\pi}{2}, \frac{\pi}{2}]$. Assuming Stokes' law is valid, what is the value of the phase constant $\phi$?

A closed circuit loop of area $A$ and total resistance $R$ contains a parallel-plate capacitor of capacitance $C$. The loop lies in the $xy$-plane and is subjected to a uniform external magnetic field $\vec{B}_{\text{ext}}(t) = B_0 \cos(\omega t) \hat{k}$. The time-varying magnetic flux induces a conduction current in the loop, which in turn creates a displacement current $I_d(t)$ between the capacitor plates. If $I_d(t)$ is expressed as $I_{\text{max}} \cos(\omega t + \phi)$ with $I_{\text{max}} > 0$, and its positive direction is defined by the right-hand rule relative to $\hat{k}$, what is the phase constant $\phi$?

A block of mass $m = 1 \text{ kg}$ is attached to a spring of force constant $k = 100 \text{ N m}^{-1}$ and undergoes weakly damped oscillations. The damping force is given by $F_d = -b(t)v$, where $v$ is the instantaneous velocity and the damping coefficient varies with time as $b(t) = b_0(1 + \alpha t)$. Given $b_0 = 0.2 \text{ kg s}^{-1}$ and $\alpha = 0.1 \text{ s}^{-1}$. Assuming the fractional loss of energy per cycle is very small, at what time $t$ will the total mechanical energy of the oscillator decrease to $1/e$ of its initial value?

A fixed point mass $M$ is located at the origin $O$. A smooth rigid circular wire of radius $R$ is fixed in a plane containing $O$, such that its center $C$ is at a distance $2R$ from $O$. A small bead of mass $m$ is released from rest at the point on the wire farthest from $O$. When it reaches the point on the wire closest to $O$, it undergoes a perfectly inelastic collision with another stationary bead of mass $m$. Find the magnitude of the normal force exerted by the wire on the combined mass immediately after the collision. (Assume $G$ is the universal gravitational constant)

A block of mass $m = 1 \text{ kg}$ is attached to a fixed wall by a light spring of force constant $k = 100 \text{ N/m}$. The block rests on a horizontal surface. The natural length of the spring is at $x = 0$. The surface is perfectly smooth for $x \ge 0.16 \text{ m}$ and rough for $x < 0.16 \text{ m}$ with a coefficient of kinetic friction $\mu_k = 0.2$. A stationary block of mass $M = 3 \text{ kg}$ is placed exactly at the boundary $x = 0.16 \text{ m}$. Block $m$ is pulled to $x = 0.20 \text{ m}$ and released from rest. When it reaches $x = 0.16 \text{ m}$, it undergoes a perfectly inelastic collision with block $M$, and they stick together. Find the maximum compression of the spring during the subsequent motion. (Take $g = 10 \text{ m/s}^2$)

A non-sinusoidal periodic voltage $V(t) = 100 \sin(\omega t) + 80 \sin(2\omega t)$ (in Volts) is applied across a series LCR circuit. At the fundamental angular frequency $\omega$, the circuit components have the following values: Resistance $R = 8 \, \Omega$, Inductive reactance $X_L = 2 \, \Omega$, and Capacitive reactance $X_C = 8 \, \Omega$. Determine the root-mean-square current $I_{\text{rms}}$ flowing through the circuit and the total average power $P$ dissipated.

A classical non-ideal gas of molar mass $M$ is in thermal equilibrium at temperature $T$ inside a spherical container of radius $R_0$. The gas is subjected to an external central potential $U(r) = \alpha r^2$ (where $\alpha > 0$). A small sensor on the inner wall of the container records the instantaneous velocity of every molecule that collides with it. What is the root-mean-square (RMS) speed of the molecules recorded by the sensor? (Assume classical statistical mechanics applies and $R$ is the universal gas constant).

A monochromatic light beam of frequency $\nu$ illuminates a photoelectric surface with work function $\phi$ ($h\nu > \phi$). The intensity of the beam is modulated such that the instantaneous photon flux (number of photons per unit area per unit time) is given by $n(t) = n_0 (1 + \cos \omega t)^2$, where $n_0$ and $\omega$ are positive constants. Assume that the photoelectric emission is instantaneous and the saturation photocurrent is strictly proportional to the incident photon flux. If $I_{avg}$ is the time-averaged saturation photocurrent over a long period, which of the following correctly gives the root-mean-square (RMS) saturation photocurrent $I_{rms}$ and the stopping potential $V_s$?

A radioactive sample contains $N_0$ active nuclei at time $t=0$. The sample has a half-life of $T_{1/2}$. The root mean square (RMS) lifetime of the nuclei, denoted by $t_{\text{rms}}$, is defined as the square root of the average of the squared lifetimes of all individual nuclei. Which of the following gives the correct expression for $t_{\text{rms}}$?

A particle of mass $m = 0.1 \text{ kg}$ is subjected to a net restoring force such that its displacement $x$ (in meters) from the origin is the superposition of two collinear simple harmonic motions given by $x_1(t) = 2 \sin(10 t)$ and $x_2(t) = 4 \sin(10 t + \frac{\pi}{3})$, where $t$ is in seconds. What is the time-averaged potential energy of the particle over a large number of oscillations?

An electric dipole of dipole moment $p$ and moment of inertia $I$ undergoes small angular oscillations in a uniform electric field $E$. The surrounding medium exerts a viscous drag torque $\tau = -b \dot{\theta}$, where $\dot{\theta}$ is the angular velocity and $b$ is a positive constant. Let $A(t)$ denote the area of the phase space ellipse $(\theta, L_\theta)$ corresponding to the instantaneous total mechanical energy of the oscillator. Assuming weak damping ($b \ll \sqrt{pEI}$), which of the following expressions gives $A(t)$ in terms of its initial value $A_0$?

A physical pendulum consists of a massless rigid rod of length $L$ pivoted freely at one end $O$. A thin straight rod (acting as the pendulum bob) of length $L$ is rigidly attached to the other end of the massless rod, such that the entire assembly forms a straight line of total length $2L$. The linear mass density of the attached bob varies as $\lambda(x) = c x$, where $x$ is the distance from the pivot $O$ and $c$ is a positive constant. If the pendulum undergoes small oscillations in a vertical plane, what is the time period of the oscillations?

A simple pendulum consists of a bob of mass $m$, density $\rho$, and charge $q$, suspended by a light string of length $L$. The pendulum is located in a mine at a depth $h = R/2$ below the Earth's surface (where $R$ is the radius of the Earth and $g$ is the surface gravity). The pendulum oscillates between two large, isolated, vertical parallel plates carrying charges $+Q$ and $-Q$. In this setup, the pendulum's mean position makes an angle $\theta_1$ with the vertical, where $\tan \theta_1 = 4/3$. A non-viscous liquid dielectric of dielectric constant $K = 5/3$ and density $\rho_L = \rho/5$ is then poured into the space between the plates, completely submerging the pendulum. Ignoring viscous drag and hydrodynamic added-mass effects, what is the new time period $T_2$ of small oscillations of the pendulum?

A torsion pendulum consists of a uniform solid disc of mass $M$ and linear expansion coefficient $\alpha_d$, suspended at its center by a uniform wire of length $L$, radius $r$, and linear expansion coefficient $\alpha_w$. The shear modulus of the wire material varies with temperature as $\eta = \eta_0 (1 - \gamma \Delta T)$, where $\gamma$ is a positive constant and $\Delta T$ is the rise in temperature. If the angular frequency of torsional oscillations remains unchanged for small variations in temperature, which of the following relations is correct?

A small magnetic dipole of magnetic moment $M$ and moment of inertia $J$ is suspended by a torsional fiber of torsional constant $C$. The equilibrium position of the dipole is aligned with the $x$-axis. A long straight wire parallel to the $y$-axis is located at $z = -d$ and carries a constant current $I_0$ in the $+y$ direction. A fixed circular conducting loop of radius $a$ and resistance $R$ is placed in the $x$-$y$ plane, centered at the origin. The dipole is given a small angular displacement $\theta$ in the $x$-$z$ plane and released. Assuming weak damping, what is the number of oscillations $N$ (where $N = \frac{\omega t}{2\pi}$) after which the amplitude of oscillation reduces to $1/e$ of its initial value?

A rigid insulating rod of length $L = 0.5 \text{ m}$ and mass $M$ has a charge $Q = 3 \mu\text{C}$ uniformly distributed along its length. The rod is pivoted at one end at the origin and is free to rotate in the $xy$-plane. The system is placed in a conservative electrostatic field whose potential is given by $V(x,y) = -E_0 \frac{x^2 - y^2}{L}$, where $E_0 = 2 \times 10^4 \text{ V/m}$. The rod is initially at rest at its stable equilibrium position. An external agent slowly rotates the rod to an angle $\theta = \pi/4 \text{ rad}$ from the equilibrium position. Calculate the work done by the external agent.

A uniform cylindrical wire of initial length $L_0$ and radius $r_0$ has a shear modulus $\eta$ and Poisson's ratio $\nu$. A heavy disc of moment of inertia $I$ is attached to its lower end, causing a small longitudinal strain $\epsilon$ (where $\epsilon \ll 1$) in the equilibrium position. The disc is then given a small angular twist and released, executing torsional oscillations. Let $T_0$ be the hypothetical time period of torsional oscillations if the wire were completely un-stretched, and $T$ be the actual time period under the load. The fractional change in the time period, $\frac{T - T_0}{T_0}$, is approximately:

A block of mass $m \text{ (kg)}$ is attached to a spring of constant $k \text{ (N/m)}$ and oscillates horizontally in a thermally insulated container filled with a viscous liquid. The total heat capacity of the block and the liquid is $C \text{ (J/K)}$. The damping force on the block is $F = -b v$, where $v \text{ (m/s)}$ is the velocity. The damping coefficient $b \text{ (kg/s)}$ depends on the temperature increase $\Delta T \text{ (K)}$ of the system as $b = b_0 (1 + \alpha \Delta T)$, where $b_0 \text{ (kg/s)}$ and $\alpha \text{ (K}^{-1}\text{)}$ are positive constants. Assume all dissipated mechanical energy is entirely converted into internal energy, raising the system's temperature uniformly. The initial mechanical energy is $E_0 \text{ (J)}$. Given that $\alpha E_0 = C$, find the time $t \text{ (s)}$ taken for the mechanical energy to reduce to $E_0 / 2$. Assume the oscillator is lightly damped such that the average rate of energy loss over a cycle is $\langle \frac{dE}{dt} \rangle = -\frac{b}{m} E$.

An ideal primary LC circuit consists of a capacitor of capacitance $C$ and an inductor of self-inductance $L_1$. It is magnetically coupled to a stationary secondary closed loop having self-inductance $L_2$ and resistance $R$. The mutual inductance between the inductor $L_1$ and the secondary loop is $M$. The primary circuit is set into oscillations. Assuming the coupling is weak ($M \ll \sqrt{L_1 L_2}$) such that the oscillation frequency is $\omega \approx \frac{1}{\sqrt{L_1 C}}$ and the primary's effective inductance is approximately $L_1$, find the damping coefficient $\alpha$ of the primary circuit's current amplitude (which decays as $e^{-\alpha t}$).

A block of mass $m$ is placed on a smooth horizontal floor of a carriage and attached to a vertical wall of the carriage via a spring of force constant $k$ and a dashpot of damping coefficient $b$. The carriage moves along the horizontal x-axis with an acceleration $a(t) = \alpha t$ for $t \ge 0$, where $\alpha$ is a positive constant. Let $x(t)$ be the displacement of the block relative to the carriage from the unstretched position of the spring. As $t \to \infty$, the phase space trajectory $(x(t), v(t))$ of the block relative to the carriage converges to a shifting attractor. Identify the coordinates of this attractor $(x_{ss}(t), v_{ss}(t))$.

A rigid square conducting loop of side length $a$, mass $m$, self-inductance $L$, and a series capacitance $C$ is attached to a horizontal spring of spring constant $k$. The loop is constrained to oscillate along the $x$-axis on a frictionless surface. A static magnetic field is applied perpendicular to the plane of the loop, given by $\vec{B} = (B_0 + \alpha x)\hat{k}$, where $B_0$ and $\alpha$ are positive constants. The total electrical resistance of the loop, including dissipative radiation resistance, is $R$. An external mechanical force $F(t) = F_0 \cos(\omega t)$ drives the loop along the $x$-axis. If the driving frequency is tuned to $\omega = \sqrt{\frac{k}{m}} = \frac{1}{\sqrt{LC}}$, what is the steady-state amplitude $X_0$ of the loop's mechanical oscillation? Assume the loop remains in the $xy$-plane and its sides remain parallel to the axes.

In a conductor, free electrons of mass $m$, charge $-e$, and number density $n$ are subjected to a high-frequency alternating electric field $E(t) = E_0 \cos(\omega t)$. The electrons experience a resistive drag force $\vec{F}_d = -m\vec{v}/\tau$, where $\vec{v}$ is the drift velocity and $\tau$ is the mean relaxation time. If the driving frequency $\omega$ is exactly equal to the collision frequency $\tau^{-1}$, what is the time-averaged power dissipated per unit volume $\langle P \rangle$ and the phase angle $\phi$ by which the current density lags the electric field?

A transverse wave propagates on a string of linear mass density $\rho$ and tension $T$. The string is embedded in a non-linear elastic medium, such that the potential energy density (energy per unit length) of the medium is given by $u_p = \frac{1}{2}\rho \omega_0^2 y^2 + \alpha y^3$, where $y(x,t)$ is the transverse displacement. Two independent wave sources emit waves which, if the medium were purely linear ($\alpha=0$), would result in the steady-state superposition $y_{lin}(x,t) = A \sin(kx - \omega t) + A \sin(kx - \omega t + \pi/3)$. Due to the non-linear term, the actual displacement $y(x,t)$ contains a local time-independent shift $y_0$. Assuming $\alpha$ is small, what is the magnitude of the constant shift $y_0$ to leading order in $A$?

A particle of mass $m = 1 \text{ kg}$ is attached to a fixed support by a spring of force constant $k = 25 \text{ N/m}$. The particle is subjected to a linear drag force $F_d = -bv$, where $b = 4 \text{ kg/s}$ and $v$ is the velocity of the particle. An external driving force $F(t) = 40 \cos(5t) + 40 \cos(3t)$ (in Newtons, with $t$ in seconds) is applied to the particle along the direction of the spring. After a sufficiently long time, the system reaches a steady state. What is the average rate of energy dissipation by the drag force over a long time interval?

A particle of mass $m$ and charge $q$ is initially at rest at the origin $(0,0)$ of a Cartesian coordinate system at time $t=0$. It is placed in a region where a quasi-static, spatially uniform but time-varying electric field is present, given by: $\vec{E}(t) = E_0 \cos(\omega t) \hat{i} + E_0 \cos(2\omega t) \hat{j}$ where $E_0$ and $\omega$ are positive constants. Which of the following represents the equation of the trajectory of the particle in the $xy$-plane?

A damped harmonic oscillator of mass $m = 1 \text{ kg}$ is subjected to a restoring force $F_k = -100x \text{ N}$ and a damping force $F_d = -2v \text{ N}$, where $x$ is the displacement in meters and $v$ is the velocity in m/s. The oscillator is driven by an external force $F(t) = F_0 (\cos(10t) + \cos(8t)) \text{ N}$, where $t$ is in seconds. In the steady state, the displacement of the particle is given by the superposition $x(t) = A_1 \cos(10t - \phi_1) + A_2 \cos(8t - \phi_2)$. What is the value of $\tan(|\phi_1 - \phi_2|)$?

A rigid, thin rod of length $L$ is suspended from a frictionless pivot at one of its ends ($x=0$). The linear mass density of the rod varies along its length as $\lambda(x) = \alpha x^2$, where $\alpha$ is a positive constant and $x$ is the distance from the pivot. If the rod is displaced by a small angle $\theta$ from the vertical and released, what is the time period $T$ of its resulting simple harmonic motion? (Assume acceleration due to gravity is $g$ and air resistance is negligible).

A particle of mass $m$ is constrained to move along the positive x-axis. Its potential energy is given by $U(x) = U_0 \left[ \frac{1}{3} \left(\frac{a}{x}\right)^3 - \frac{1}{2} \left(\frac{a}{x}\right)^2 \right]$ where $U_0$ and $a$ are positive constants. The particle is initially at rest at its stable equilibrium position. A short impulse $J$ is delivered to the particle in the positive x-direction. Given that $J \ll \sqrt{m U_0}$, the maximum x-coordinate reached by the particle is approximately:

A uniform massless elastic rod of 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 system is initially in equilibrium at a temperature $T_0$. The temperature of the rod is then uniformly increased to $T_0 + \Delta T$. Let $\alpha$ be the coefficient of linear expansion of the material of the rod. Assuming the cross-sectional area $A$ and Young's modulus $Y$ remain constant with temperature, and $g$ is the acceleration due to gravity, which of the following expressions correctly gives the downward shift in the equilibrium position of the block ($\Delta y$) and the new angular frequency of small vertical oscillations ($\omega'$) at the higher temperature?

A rigid circular loop of radius $R = 0.1 \text{ m}$ and mass $m = 0.5 \text{ kg}$ is connected to a central hub by massless rigid spokes. The hub is suspended by a massless vertical spring of spring constant $k = 16 \text{ N/m}$, constraining the loop to move only along the vertical $z$-axis. The system is placed in an axisymmetric magnetic field given by $\vec{B} = -\alpha z r \hat{r} + (B_0 + \alpha z^2) \hat{z}$, where $r$ is the radial distance from the $z$-axis, $\hat{r}$ and $\hat{z}$ are unit vectors in cylindrical coordinates, $B_0 = 1.0 \text{ T}$, and $\alpha = \frac{45}{\pi} \text{ T/m}^2$. The equilibrium position of the loop in the absence of any current is at $z = 0$. A steady current $I = 10 \text{ A}$ is then established in the loop, flowing in the clockwise direction when viewed from the positive $z$-axis. If the loop is given a small vertical displacement from its equilibrium position, what is the period of its simple harmonic oscillations?

A light spring of force constant $k = 100 \text{ N/m}$ hangs vertically from a fixed support. A small ring of mass $m = 1.0 \text{ kg}$ is attached to the lower end of the spring and is threaded on a rough vertical guide rod. A constant horizontal force $F = 20 \text{ N}$ presses the ring against the rod. The coefficient of kinetic friction between the ring and the rod is $\mu_k = 0.20$, and the coefficient of static friction is $\mu_s = 0.30$. The ring is pulled downwards to a distance of $39 \text{ cm}$ from the relaxed position of the spring and released from rest. Taking $g = 10 \text{ m/s}^2$, what is the total distance traveled by the ring before it comes to a permanent stop?

A horizontal conducting rod of mass $m = 0.1 \text{ kg}$ and length $l = 1 \text{ m}$ is suspended by a vertical ideal spring of spring constant $k = 6 \text{ N/m}$. The rod is placed in a uniform horizontal magnetic field $B = 0.8 \text{ T}$ perpendicular to its length and the direction of motion. The ends of the rod are connected to a stationary external circuit containing a resistor $R = 1 \ \Omega$ and an inductor $L = 0.1 \text{ H}$ in series. The system is set into small vertical oscillations. The vertical displacement $x(t)$ of the rod can be expressed as a sum of a purely exponentially decaying term and an exponentially decaying oscillatory term. What is the time constant $\tau$ for the decay of the amplitude of the oscillatory term? (The amplitude of the oscillatory term decays as $A_0 e^{-t/\tau}$).

A physical pendulum consists of a uniform solid metal disk of mass $M$ (in kg) and radius $R$ (in m). A massless rigid rod of length $L = R$ (in m) is attached to the edge of the disk such that it extends radially outward. The pendulum is pivoted at the free end of the rod and oscillates in a vertical plane with a small angular amplitude. The rod is made of INVAR, which has a negligible coefficient of thermal expansion ($\alpha_{\text{invar}} \approx 0$), while the disk is made of a metal with a coefficient of linear thermal expansion $\alpha$ (in K$^{-1}$). If the temperature of the entire system is increased by a small amount $\Delta \theta$ (in K), what is the fractional change in the time period of oscillation, $\frac{\Delta T}{T}$? (Assume $\alpha \Delta \theta \ll 1$)

A simple pendulum consists of a small bob of mass $m$ attached to a rigid, massless rod of length $L$. The other end of the rod is rigidly attached to a cylindrical shaft of radius $r$ ($r \ll L$) and negligible mass. The shaft rotates in a fixed, horizontal dry bearing with a coefficient of kinetic friction $\mu$. The pendulum is swinging upwards, and at the instant it makes an angle $\theta$ with the downward vertical, its angular speed is $\omega$. Assume that the normal force exerted by the bearing on the shaft is equal to the radial tension in the rod. Which of the following expressions correctly represents the magnitude of the angular acceleration $\alpha$ of the pendulum at this instant?

A rigid superconducting square loop of mass $m$, side length $L$, and self-inductance $L_0$ is hinged freely at its top horizontal edge. It hangs as a physical pendulum in a uniform downward vertical magnetic field $\vec{B} = B_0 \hat{j}$. The acceleration due to gravity is $g$ (downward). At equilibrium ($\theta = 0$), the loop lies in the vertical plane and the current in the loop is zero. If the loop is given a small angular displacement $\theta$ and released, what is the angular frequency $\omega$ of the resulting simple harmonic motion? (Assume the wires of the loop are thin and uniform).

A uniform square plate of mass $M$ and side length $a$ is suspended from a frictionless pivot at one of its corners. The plate lies in a vertical plane and is free to oscillate under gravity. It is displaced by a small angle $\theta_0$ (in radians) from its stable equilibrium position and released from rest. The gravitational restoring torque can be modeled as an effective torsional spring with a rotational spring constant $k_{eff}$. Assuming the acceleration due to gravity is $g$, what is the value of $k_{eff}$ and the maximum angular velocity $\omega_{max}$ of the plate?

A uniform rigid rod of mass $m$ and length $L$ is suspended from a frictionless hinge attached to the ceiling of a train compartment. The train moves along a straight horizontal track with a constant acceleration $a = \sqrt{3}g$, where $g$ is the acceleration due to gravity. The rod is allowed to reach its stable equilibrium position relative to the train. If the rod is given a small angular displacement in the plane of acceleration, what is the time period $T$ of its small oscillations?

An ideal capacitor of capacitance $C$ (in Farads) is initially charged to a charge $Q_0$ (in Coulombs). At $t=0$, it is connected in series with an inductor of inductance $L$ (in Henrys) and a non-linear resistor. The non-linear resistor has a voltage-current relationship given by $V_R = \alpha I^2$, where $I$ is the magnitude of the current (in Amperes) and $\alpha$ is a positive constant (in $\text{V}\cdot\text{A}^{-2}$). Assuming the resistance opposes the flow of current, find the maximum current $I_{max}$ in the circuit during the first discharge phase.

A horizontal cylinder of cross-sectional area $A$ contains an ideal monoatomic gas at an initial equilibrium pressure $P_0$ and volume $V_0$. A movable piston of mass $m$ seals the gas and is connected to a fixed vertical wall by a linear spring of spring constant $k$. The region on the other side of the piston is a vacuum. At equilibrium, the spring is in a stretched state. An external agent slowly displaces the piston outward by a small distance $x$ (where $x \ll V_0/A$). Let $W_a$ be the total work done by the external agent if the gas undergoes a reversible adiabatic process, and $W_i$ be the total work done if the gas undergoes a reversible isothermal process. Which of the following expressions correctly represents the difference $(W_a - W_i)$ to the leading order in $x$?

A uniform vertical U-tube of inner radius $r$ contains a liquid of density $\rho$ and total length $L$. The inner surface of the tube has a special coating such that the contact angle $\theta$ of the liquid with the tube wall varies with the vertical height $y$ from the horizontal base of the U-tube according to the relation $\cos\theta = \cos\theta_0 - \alpha y$, where $\alpha$ is a positive constant and $\theta_0$ is the contact angle at the base. The surface tension of the liquid is $T$ and the acceleration due to gravity is $g$. The liquid column is given a small displacement and released. Assuming the liquid always wets the tube ($\theta < 90^\circ$) and ignoring viscous damping, what is the angular frequency $\omega$ of the resulting small oscillations?

A mechanical oscillator (mass $M = 3 \text{ kg}$, spring constant $K = 100 \text{ N/m}$) is modeled electrically using the Force-Voltage analogy, where velocity maps to current. The resulting equivalent ideal LC tank circuit is connected in series with a physical RL coil having inductance $L_0 = 1 \text{ H}$ and resistance $R_0 = 8 \text{ }\Omega$.

At $t = 0$, with the circuit initially relaxed (zero current, zero charge), an impulsive voltage $V(t)$ is applied across the entire series combination such that $\int_{0^-}^{0^+} V(t) dt = 24 \text{ V}\cdot\text{s}$. Which of the following expressions gives the charge $q(t)$ on the capacitor for $t > 0$?

A thin bimetallic strip of length $L$ and constant flexural rigidity $EI$ is clamped at the origin. Its tangent at the origin points along the positive $x$-axis. A point mass $m$ is attached to its free end. At temperature $T_0$, the strip is perfectly straight. When heated to $T_0 + \Delta T$, thermal stresses cause the strip to bend into a perfect semicircle in the upper half of the $xy$-plane ($y \ge 0$). A rigid, massless guide rod is pivoted at the origin and attached to the mass $m$, constraining it to move only along a circular path centered at the origin. The mass $m$ is given a small angular displacement and undergoes rotational oscillations. Assuming the bimetallic strip provides the only restoring force, what is the angular frequency $\omega$ of these oscillations?

A block of mass $m = 2.0 \text{ kg}$ is placed on a rough horizontal surface and connected to a vertical wall via two massless springs in series. The spring constants are $k_1 = 300 \text{ N/m}$ and $k_2 = 600 \text{ N/m}$. The coefficients of static and kinetic friction between the block and the surface are $\mu_s = 0.25$ and $\mu_k = 0.20$, respectively. The block is pulled to a distance $x_0 = 15.0 \text{ cm}$ from its equilibrium (unstretched) position and released from rest. Taking $g = 10 \text{ m/s}^2$, calculate the total distance traveled by the block before it comes to a permanent rest.

A torsional pendulum consists of a disc of moment of inertia $I \text{ (kg m}^2\text{)}$ and a suspension wire of torsional constant $C \text{ (N m rad}^{-1}\text{)}$. A closed-loop coil of $N$ turns and area $A \text{ (m}^2\text{)}$ is rigidly attached to the disc and placed in a uniform radial magnetic field $B \text{ (T)}$. The coil has an initial electrical resistance $R_0 \text{ (}\Omega\text{)}$ at temperature $T_0 \text{ (K)}$, a temperature coefficient of resistance $\alpha \text{ (K}^{-1}\text{)}$, and a heat capacity $H \text{ (J K}^{-1}\text{)}$. The pendulum is given an initial angular displacement $\theta_0 \text{ (rad)}$ and released. The damping is purely electromagnetic. Assuming all dissipated mechanical energy is converted into heat within the coil with no heat loss to the surroundings, what is the final quality factor $Q_f$ of the system after the oscillations have completely damped out? (Given the initial quality factor is $Q_0$)

An AC voltage source $V(t) = V_0 \sin(\omega t)$ is connected in series with a resistor of resistance $R$, an inductor of inductance $L$, and a parallel plate capacitor. The capacitor is initially filled with a solid dielectric. The dielectric is being withdrawn at a constant speed such that the capacitance decreases linearly with time as $C(t) = C_0 - \beta t$, where $\beta > 0$ is a constant rate in $\text{F/s}$. Assume that the observation time is sufficiently small such that $\beta t \ll C_0$, allowing $C(t)$ to be approximated as $C_0$ in its magnitude, but its rate of change $\frac{dC}{dt} = -\beta$ remains significant. Determine the phase angle $\phi$ by which the source voltage leads the current in the pseudo-steady state.

Assertion (A): A composite solid cylinder of radius $R$ and total mass $M$ is formed by taking a uniform cylinder of density $\rho$ and replacing a cylindrical portion of radius $R/2$ (centered at a distance $R/2$ from the geometric axis) with a material of density $3\rho$. When placed on a rough horizontal surface and slightly displaced from its stable equilibrium, it rolls without slipping with a time period $T = 3\pi \sqrt{\frac{3R}{g}}$, where $g$ is the acceleration due to gravity.

Reason (R): The restoring torque about the instantaneous axis of rotation is provided by the weight of the cylinder. The moment of inertia about the contact point can be calculated using the parallel axis theorem as $I_P = I_C + MR^2$, where $I_C$ is the moment of inertia about the geometric center.

A torsional pendulum consists of a uniform horizontal disk of mass $M$ (in kg) and radius $R$ (in m) suspended by a vertical wire of torsional constant $C$ (in N$\cdot$m/rad). At $t=0$, the disk is released from rest at an angular displacement $\theta_0$ (in rad). Immediately after release, sand begins to fall vertically onto the edge of the disk at a constant rate $\mu$ (in kg/s) and sticks to it. Assume $\mu$ is sufficiently small such that the adiabatic approximation holds, meaning the accumulated phase of oscillation is given by $\Phi(t) = \int_0^t \omega(t') dt'$, where $\omega(t)$ is the instantaneous natural frequency. The angular position of the disk at time $t$ can be locally expressed as $\theta(t) \approx A(t) \cos(\omega(t) t + \delta(t))$. Which of the following expressions gives the effective phase constant $\delta(\tau)$ at time $t=\tau$ (in s)?

(Let $I_0 = \frac{1}{2}MR^2$ and $I_\tau = I_0 + \mu R^2 \tau$, both in kg$\cdot$m$^2$)

Assertion (A): A bucket containing sand of total instantaneous mass $M(t)$ is suspended from a vertical spring of stiffness $k$ and oscillates. Sand leaks out from a hole at the bottom at a constant, very slow rate $\mu = -\frac{dM}{dt}$ (with zero relative velocity with respect to the bucket). During this process, the amplitude of oscillation $A$ of the bucket decreases as $A \propto (M(t))^{1/4}$.

Reason (R): The leaking sand exerts zero thrust force, yielding the equation of motion $M(t)\ddot{z} + kz = 0$ about the instantaneous mean position $z=0$. The continuous shedding of mass removes kinetic energy from the system at a rate $\frac{1}{2}\dot{M}\dot{z}^2$, causing the amplitude to decrease, whereas applying the standard adiabatic invariant $\frac{E}{\omega} = \text{constant}$ would incorrectly predict an increasing amplitude.

A particle of mass $m$ and charge $+q$ is initially at rest at the origin of a Cartesian coordinate system. At $t = 0$, it is subjected to a time-varying electric field given by $\vec{E} = E_0 \cos(\omega t) \hat{i} + E_0 \cos(2\omega t) \hat{j}$, where $E_0$ and $\omega$ are positive constants. The region is free of gravity and magnetic fields. What is the maximum kinetic energy attained by the particle during its subsequent motion?

A simple pendulum consists of a small bob of mass $m$ and positive charge $+q$ suspended by a massless, inextensible string of length $L$. The pendulum is attached to the upper plate of a horizontal parallel-plate capacitor with plate separation $d$. The capacitor is connected to an ideal battery of constant voltage $V$, such that the upper plate is maintained at a positive potential relative to the lower plate. The entire setup is located in a mine at a depth $h = R/2$ below the Earth's surface, where $R$ is the radius of the Earth and $g$ is the acceleration due to gravity at the surface. It is given that $qV = \frac{1}{2} mgd$. Initially, the space between the plates is vacuum. A solid, uncharged dielectric slab of dielectric constant $K = 3$ and thickness $d/2$ is then slowly inserted and placed on the lower plate. Assuming the pendulum bob always oscillates in the vacuum region ($L < d/2$) and edge effects are negligible, what is the new time period of small oscillations of the pendulum?