Work, Energy, and Power - Class 11 Part 1

An electric dipole of dipole moment magnitude $p$ is initially placed at the origin $O$ of a coordinate system, pointing in the $+y$-direction. It is immersed in a non-uniform electrostatic field given by $\vec{E} = \alpha (x \hat{i} - y \hat{j})$, where $\alpha$ is a positive constant. The dipole is moved to the point $A(d, d)$ and rotated to point in the $+x$-direction. Consider two distinct processes for this transition:

Process 1: The dipole is first translated from $O$ to $A$ along the line $y=x$ while maintaining its orientation in the $+y$-direction. Then, at point $A$, it is rotated in the $xy$-plane to align with the $+x$-direction. Process 2: The dipole is first rotated at $O$ in the $xy$-plane to align with the $+x$-direction. Then, it is translated from $O$ to $A$ along the line $y=x$ while maintaining this new orientation.

Let $W_{T1}$ and $W_{R1}$ be the translational and rotational work done by the electric field in Process 1, respectively. Let $W_{T2}$ and $W_{R2}$ be the corresponding works in Process 2. Which of the following options correctly gives these quantities?

An alternating voltage $V(t) = 200 \sin(\omega t) + 100 \sin(3\omega t)$ is applied across a complex circuit element. The resulting current through the circuit is measured to be $I(t) = 10 \sin(\omega t - \frac{\pi}{3}) + 5 \sin(2\omega t) + 5\sqrt{2} \sin(3\omega t - \frac{\pi}{4})$. All quantities are in SI units. What is the average power dissipated in the circuit over a long time?

A long straight wire carrying a steady current $I$ lies along the $z$-axis. A curved open surface $S$ is defined by the parabolic equation $y = \frac{x^2}{a}$ for the range $a \le x \le 2a$, and is bounded in the $z$-direction by $0 \le z \le L$. Calculate the magnitude of the magnetic flux through this surface $S$.

A small particle of mass $m$ moving with a uniform velocity $v_0 \hat{i}$ collides elastically with a smooth, free sphere of mass $M = 3m$ and radius $R$ initially at rest at the origin. The impact parameter for the collision is $b = \frac{R}{2}$, and the particle's initial y-coordinate is positive. What is the final velocity vector of the particle immediately after the collision?

An infinite non-conducting sheet lies in the $xy$-plane with a variable surface charge density $\sigma(x) = \alpha x$, where $\alpha$ is a positive constant. In the region $z > 0$, the electrostatic field $\vec{E}$ is conservative, and its $z$-component is given by $E_z = \frac{\sigma(x)}{2\epsilon_0}$. It is also given that the $x$-component of the electric field vanishes on the sheet ($E_x = 0$ at $z=0$) and the $y$-component is zero everywhere. A particle of charge $q$ is moved from the origin $(0,0,0)$ to the point $(a, 0, b)$ along the path $x = a\left(\frac{z}{b}\right)^2$. What is the work done by the electric field on the particle?

A particle of mass $m = 1 \text{ kg}$ is initially at rest at the origin of a Cartesian coordinate system. It is acted upon by a time-varying force $\vec{F}(t) = (2t)\hat{i} + (3t^2)\hat{j} \text{ N}$, where $t$ is in seconds. What is the magnitude of the angular momentum of the particle about the point $P(0, 1 \text{ m})$ at $t = 2 \text{ s}$?

One mole of an ideal gas is enclosed in a horizontal cylinder equipped with a piston of area $A$. The gas expands quasi-statically and isothermally at a constant temperature $T_0$ from an initial volume $V_0$ to $2V_0$. During the expansion, the piston moves against a rough surface, experiencing a variable frictional force $F_f = \frac{A R T_0}{2 V_0^2} (V - V_0)$, where $V$ is the instantaneous volume of the gas and $R$ is the universal gas constant. If all the heat generated by the friction is completely absorbed by the gas, what is the heat $Q_{\text{res}}$ supplied by the external thermal reservoir to maintain the isothermal process?

A particle of mass $m$ moves in one dimension $x > 0$ under the influence of a conservative force field. The potential energy of the particle is given by $U(x) = U_0 \left[ \ln\left(\frac{x}{a}\right) + \cos\left(\frac{x}{a}\right) \right]$ where $U_0$ and $a$ are positive constants. For large values of $x$ ($x \gg a$), the positions of the stable equilibrium points are approximately given by (where $n$ is a large positive integer):

A sealed horizontal tube of length $L$ is completely filled with an ideal incompressible fluid of density $\rho$. A small block of volume $V$ and density $\sigma$ (where $\sigma < \rho$) is initially at rest at the rear end of the tube. The tube is given a constant horizontal acceleration $a_0$ along its length. Ignoring viscous drag and added mass effects, what is the work done by the fluid on the block in the ground frame from the start of the motion until the block reaches the front end of the tube?

A solid sphere of mass $m$ and radius $R$ translates with a velocity $v_0$ without rotation on a smooth horizontal plane. It suddenly encounters a perfectly rough horizontal plane, causing it to instantaneously achieve pure rolling. What is the magnitude of the work done by the impulsive friction force during this instantaneous transition?

A parallel plate capacitor has square plates of side length $L$ and separation $d$. It is kept connected to a battery of constant electromotive force $V$. A dielectric slab of dielectric constant $K$ and dimensions $L \times L \times d$ is slowly and completely inserted between the plates by an external agent. What is the work done by the external agent, $W_{\text{ext}}$? (Assume $\epsilon_0$ is the permittivity of free space and neglect edge effects).

A parallel plate capacitor with plates of length $L$, width $w$, and separation $d$ is connected to a battery of constant voltage $V$. A solid dielectric slab of dielectric constant $K$ and dimensions $L \times w \times d$ is inserted a distance $x$ into the capacitor along its length. The $+x$ direction is defined along the inward direction of insertion, with unit vector $\hat{i}$. Which of the following correctly represents the electrostatic force $\vec{F}_e$ exerted by the capacitor on the slab, and the external work $W_{ext}$ required to slowly pull the slab completely out from a fully inserted state?

A small spherical particle of mass $m$ and radius $r$ is released from rest in a large column of viscous fluid. The fluid has a coefficient of viscosity $\eta$ and its density is negligible compared to the density of the particle. The particle falls under the uniform gravitational field $g$. Assuming Stokes' law is valid throughout the motion, what is the work done by the viscous drag force on the particle from the instant of release until it attains exactly half of its terminal velocity?

A solid non-conducting sphere of radius $R$ carries a uniform volume charge density $\rho$. A spherical cavity of radius $R/3$ is excavated from the sphere such that the center of the cavity is at a distance $2R/3$ from the center of the solid sphere. Let $V_O$ be the electrostatic potential at the center of the solid sphere and $V_C$ be the electrostatic potential at the center of the cavity. Assuming the potential at infinity to be zero, find the ratio $\frac{V_O}{V_C}$.

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density given by $\rho(r) = \rho_0 \frac{r}{R}$ for $r \le R$, where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. The electrostatic potential at the surface of the sphere is explicitly defined to be zero ($V(R) = 0$). Which of the following correctly gives the electrostatic potential at the center of the sphere ($V_c$) and at a radial distance $r = 2R$ ($V_{2R}$)? Let $\epsilon_0$ be the permittivity of free space.

A liquid flows steadily downwards through a vertical pipe of constant cross-sectional area $A$. Due to a steady thermal gradient, the density of the liquid varies with depth $z$ as $\rho(z) = \rho_0 (1 + \alpha z)$, where $\rho_0$ and $\alpha$ are positive constants. The mass flow rate of the liquid is $\dot{m}$. If the pressure at $z=0$ is $P_0$, what is the pressure $P_L$ at depth $z=L$?

A non-uniform spherical planet of mass $M$ and radius $R$ has a radially dependent density profile $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$, where $r$ is the distance from the center. A small payload of mass $m$ ($m \ll M$) is initially at rest at the exact center of the planet. The payload is then launched to infinity through a narrow radial shaft. Simultaneously, the planet undergoes a structural collapse under its own gravity, settling into a uniform sphere of radius $R/2$. What is the total change in the gravitational potential energy of the system (planet + payload)?

A perfectly conducting straight rod of mass $m$ and length $l$ slides without friction down two vertical, parallel conducting rails. A uniform horizontal magnetic field $B$ exists perpendicular to the plane of the rails. The top ends of the rails are connected to an ideal solenoid of inductance $L$ and internal volume $V_0$. The system is released from rest at $t=0$. Assuming the magnetic field inside the solenoid is uniform and confined to its volume $V_0$, which of the following expressions represents the time-averaged magnetic energy density $\langle u_B \rangle$ inside the solenoid over one complete period of oscillation?

A small block of mass $m$ is projected from the lowest point of a fixed, smooth vertical circular track of radius $R$. The track is immersed in a fluid that exerts a tangential drag force on the block of magnitude $F_d = \frac{m v^2}{2R}$, where $v$ is the instantaneous speed of the block. What is the minimum initial speed $v_0$ required at the lowest point for the block to just complete the vertical circle? (Assume acceleration due to gravity is $g$)

A parallel plate capacitor has rectangular plates of length $L$, width $w$, and separation $d$. It is given a charge $Q$ and then isolated. A dielectric slab of mass $m$, dielectric constant $K$, and dimensions $L \times w \times d$ is attached to a relaxed spring of spring constant $k$. The slab is partially inserted into the capacitor by a distance $x$ parallel to its length. The system reaches an equilibrium state where the slab is inserted by a distance $x_0$ ($0 < x_0 < L$). If the slab is slightly displaced from this equilibrium position, what is the angular frequency $\omega$ of the resulting small oscillations? Assume friction and edge effects are negligible.

A satellite of mass $m$ is in a bound planar orbit around a planet of mass $M$ ($M \gg m$). The satellite is connected to the center of the planet by a massless elastic tether of natural length zero and unknown spring constant $k$. The maximum and minimum distances of the satellite from the planet's center are $3R$ and $R$, respectively. If the speed of the satellite at its closest approach is $\sqrt{\frac{2GM}{R}}$, what is the radius of curvature of the satellite's trajectory at its farthest point?

A horizontal Venturi tube has a wide section of cross-sectional area $A$ (at point 1) and a throat of area $A/2$ (at point 2). An ideal, incompressible fluid of density $\rho$ flows through the tube. The volume flow rate is non-steady and given by $Q(t) = \beta t$, where $\beta$ is a positive constant. A differential pressure sensor connects point 1 and point 2. The sensor consists of a small cylinder of area $a$ and a massless piston attached to a spring of stiffness $k$. The geometric integral along the streamline from point 1 to point 2 is given as $\int_1^2 \frac{dx}{A(x)} = \frac{L}{A}$. Assuming the volume of fluid drawn by the sensor is negligible and the piston adjusts instantaneously, what is the compression $x$ of the spring at time $t$?

A thin cylindrical rod of length $L$ and uniform cross-sectional area $A$ has a non-uniform density given by $\rho(x) = \rho_0 \left(1 + \frac{x}{L}\right)$, where $x$ is the distance from one end ($x=0$). The rod is completely submerged in a large tank of ideal liquid of uniform density $\rho_0$. The rod is initially held at rest in a horizontal position at a sufficient depth and then released. Assuming viscous drag is negligible, find the angular velocity $\omega$ of the rod when it reaches the vertical position for the first time.

A parallel plate capacitor with plates of width $b$, length $L$, and separation $d$ is connected to a constant voltage source $V$. A dielectric slab of mass $m$, dielectric constant $K$, and dimensions matching the capacitor's interior is initially placed just at the edge of the plates ($x=0$). The slab is pulled into the capacitor by electrostatic forces. As it enters, it experiences a position-dependent resistive friction force $f = \alpha x$, where $\alpha$ is a positive constant and $x$ is the length of the slab inside the capacitor. Assuming gravity is negligible and the plates are long enough, what is the maximum kinetic energy acquired by the dielectric slab?

A horizontal, perfectly insulated cylinder is placed in a vacuum and contains $1 \text{ mol}$ of a diatomic gas at an initial temperature $T_0$. A frictionless, adiabatic piston of mass $M$ is given an initial inward velocity $v_0$ such that $M v_0^2 = 18 R T_0$, compressing the gas. Assuming the gas maintains internal thermal equilibrium, and due to the excitation of vibrational degrees of freedom at higher energies, its molar heat capacity at constant volume varies as $C_V(T) = \frac{5}{2}R + 2R\left(\frac{T - T_0}{T_0}\right)$ for $T \ge T_0$. What is the maximum temperature $T_{max}$ reached by the gas?

A small horizontal circular metallic loop of radius $r$, mass $m$, and resistance $R$ is released from rest at $z=0$ and falls under gravity (acceleration $g$) along the positive z-axis (downwards). The magnetic field in the region is axisymmetric and its vertical component is given by $B_z = B_0 (1 + \alpha z)$, where $B_0$ and $\alpha$ are positive constants. The loop reaches a terminal velocity well before it has fallen a distance $H$. Assuming the loop remains horizontal and its radius $r$ is small enough that the magnetic field components can be approximated to first order in $r$, what is the total electrical energy dissipated as Joule heating in the loop during its fall from $z=0$ to $z=H$?

A point charge $+q$ is initially placed at the exact center of a fixed, isolated, uncharged thin conducting spherical shell of inner radius $R$. The charge is then displaced by a very small vector $\vec{x}$ ($|\vec{x}| \ll R$) from the center. Which of the following represents the approximate electrostatic force $\vec{F}$ acting on the point charge at this displaced position?

A horizontal tube of moment of inertia $I_0$ rotates freely about a vertical axis passing through its center. A particle of mass $m$ can slide without friction inside the tube. The particle is connected to the axis of rotation by a light spring of spring constant $k$ and natural length zero. The total angular momentum of the system is $L$ and is strictly conserved. The particle is in a stable dynamic equilibrium at a radial distance $r_0 > 0$. What is the angular frequency $\Omega$ of small radial oscillations of the particle about $r_0$?

A perfectly insulated vertical cylinder of cross-sectional area $A$ contains an ideal gas with adiabatic exponent $\gamma$. A perfectly insulating, frictionless piston of mass $M$ rests in equilibrium at a height $L$ above the base. The external atmospheric pressure is $P_0$. The piston is given a small vertical displacement $x$ ($x \ll L$). The total potential energy of the system (gas, piston, and atmosphere) near equilibrium is expressed as $U(x) \approx U_0 + \frac{1}{2} k_{\text{eff}} x^2$. What is the effective stiffness $k_{\text{eff}}$ of the system?

A particle of mass $m = 2 \text{ kg}$ is bound in a one-dimensional potential $U(x) = \frac{1}{2} m \omega_0^2 x^2$ with natural frequency $\omega_0 = 5 \text{ rad/s}$. It experiences a viscous drag force $F_d = -b v$, where $b = 6 \text{ kg/s}$ and $v$ is its velocity. A transverse wave $y(z,t) = A \cos(kz - \omega t)$ with amplitude $A = 0.5 \text{ m}$ and frequency $\omega = 4 \text{ rad/s}$ propagates on a nearby string of linear mass density $\mu = 0.2 \text{ kg/m}$. The wave couples to the particle, exerting a driving force $F(t) = C \langle u \rangle \cos(\omega t)$, where $\langle u \rangle$ is the time-averaged energy density of the wave and $C = 50$ is a dimensionless coupling constant. In the steady state, what is the magnitude of the net force acting on the particle at the exact instant the driving force reaches its positive maximum?

A particle of mass $m = 3 \text{ kg}$ moving with an initial velocity $v_0 = 6 \text{ m/s}$ along the positive $x$-axis collides with a specialized energy-absorbing surface at $x = 0$. During the penetration ($x > 0$), the surface exerts a conservative restoring force derived from the potential $U(x) = A x^3$, where $A = 5 \text{ N/m}^2$. Additionally, the surface material exerts a non-conservative dissipative force of magnitude $F_d = B x^2$, where $B = 12 \text{ N/m}^2$, which always opposes the particle's velocity. What is the magnitude of the total impulse delivered by the particle to the surface during the entire collision process?

A particle of mass $m = 2 \text{ kg}$ is constrained to move along the vertical $y$-axis. The effective potential energy of the system (including gravity) is given by $U(y) = 3y^4 - 28y^3 + 84y^2 - 96y$ (in Joules, where $y$ is in meters). The particle is initially at rest at its global stable equilibrium position. An impulsive force is applied to the particle, giving it just enough energy to cross the potential barrier and enter the meta-stable equilibrium well. Determine the minimum impulse $J_{\text{min}}$ required to achieve this transition, and the angular frequency $\omega'$ of small oscillations if the particle were to settle at this new meta-stable equilibrium.

A rigid, closed conducting loop of arbitrary shape has mass $m$, enclosed area $A$, and electrical resistance $R$. It is constrained to translate without rotation on a smooth horizontal $xy$-plane. A non-uniform vertical magnetic field $\vec{B} = \alpha x \hat{k}$ exists in the region, where $\alpha$ is a positive constant. At $t=0$, the loop is at rest and is subjected to an instantaneous mechanical impulse $J$ in the positive $x$-direction. Neglecting self-inductance, what is the total distance $\Delta x$ traveled by the loop before it comes to rest?

A non-uniform rod of mass $M$ and length $L$ lies at rest on a smooth horizontal surface. Its linear mass density varies as $\lambda = kx$, where $x$ is the distance from end A. A small particle of mass $M$ moving with velocity $v_0$ perpendicular to the rod strikes it at end B ($x=L$). If the coefficient of restitution for the collision is $e = \frac{1}{3}$, find the velocity of the particle immediately after the collision.

A small block of mass $m$ slides with speed $v_0$ along a smooth horizontal platform. The platform ends at point $A$, where the block enters the inside of a fixed vertical circular track of radius $R$. The center $O$ of the track is at a height $R/2$ above the platform, and the block moves downwards along the track after entry. The track is smooth everywhere except at $A$, where there is a small rough patch with a coefficient of friction $\mu = \frac{1}{3\sqrt{3}}$. The collision of the block with the track at $A$ is perfectly inelastic in the normal direction. Find the minimum value of $v_0$ required for the block to complete the vertical circle without losing contact with the track.

A small incompressible solid sphere of mass $m$ and density $\rho$ is released from rest just below the free surface of a pool of liquid with density $\sigma$ ($\sigma < \rho$) and depth $H$. The atmospheric pressure at the free surface is $P_0$. The liquid exerts a drag force of magnitude $F_d = k v^2$ on the sphere, where $v$ is its speed. The sphere reaches a terminal speed $v_0$ well before striking the horizontal bottom. The collision with the bottom has a coefficient of restitution $e$. Assuming the collision time is infinitesimal, what is the maximum height $h$ the sphere rises above the bottom after the first bounce?

A uniform rigid rod of mass $M = 4m$ and length $L$ rests on a smooth horizontal surface. A small ball of mass $m$ moving with a velocity $v_0$ perpendicular to the rod strikes it elastically ($e=1$) at a distance $d = L/4$ from its center of mass. Which of the following represents the velocity of the ball immediately after the collision? (Assume the standard sign convention where the initial velocity direction is positive).

A fixed solid conducting sphere of radius $R$ carries a total charge $+Q$. A smooth, rigid, insulating circular wire of radius $a$ ($a > R$) is fixed in the vertical $x-y$ plane, concentric with the sphere. The $+y$-axis points vertically upwards. A small bead of mass $m$ and charge $+q$ is threaded on the wire. A uniform magnetic field $\vec{B} = -B_0 \hat{k}$ exists throughout the space. The bead is given a negligible push in the $+x$ direction from the highest point of the wire. What is the magnitude of the normal force exerted by the wire on the bead when it reaches the lowest point? (Assume $k = \frac{1}{4\pi\epsilon_0}$ and acceleration due to gravity is $g$ in the $-y$ direction)

An electron (charge $-e$, mass $m$) is in a circular orbit of radius $R_0$ in the $xy$-plane under the influence of a spatially varying, time-dependent magnetic field $\mathbf{B}(r, t) = B(t) \left( 1 - \alpha r^2 \right) \hat{k}$, where $r$ is the distance from the origin. The field strength parameter $B(t)$ is slowly increased over time. It is observed that the orbital radius of the electron remains strictly constant at $R_0$ during this entire process. What is the value of the constant $\alpha$?

A small sphere of volume $V$ and density $3\rho$ is attached to a fixed center by a massless inextensible string of length $R$ and moves inside a vertical circular tube of mean radius $R$. The tube contains an incompressible ideal fluid of density $\rho$ flowing steadily. The tube's cross-section varies such that the fluid velocity is $v_f(\theta) = v_0 \sqrt{\cos\theta}$ for $\theta \in [0, \pi/2]$, where $\theta$ is the angle measured from the lowest point. Due to the drag force from the fluid, the sphere moves at a constant speed $v_s = \sqrt{2gR}$. Given $v_0 = 2\sqrt{gR}$, let $T$ be the string tension at $\theta = \pi/3$, and $W_D$ be the magnitude of the work done by the drag force on the sphere from $\theta = 0$ to $\theta = \pi/3$. Find the ratio $\frac{T}{W_D/R}$.

A simple pendulum consists of a bob of mass $m$ and a light string of measured length $R$. It is released from rest at a small angular displacement $\theta_0$. At its lowest point, it undergoes a perfectly elastic head-on collision with a stationary identical bob of mass $m$, which is suspended by a light string of measured length $L$. The angle $\theta_0$ is calculated using the measured lengths and set exactly such that the second bob would just complete a vertical circular motion. If the measured lengths $R$ and $L$ have maximum percentage errors of $2\%$ and $4\%$ respectively compared to their actual physical values, and $m$ and $g$ are exact, what is the maximum percentage error in the actual tension $T$ experienced by the second string immediately after the collision? (Assume the small angle approximation is valid).

A solid sphere of mass $m$ and radius $R$ is given an initial angular velocity $\omega_0$ about its horizontal axis and is gently placed on a rough horizontal surface. The initial translational velocity of the center of mass is zero. If the coefficient of kinetic friction between the sphere and the surface is $\mu$, what is the total work done by the frictional force on the sphere from the moment it is placed on the surface until it begins pure rolling?

A block of mass $m = 9 \text{ kg}$ is placed on a rough wedge of angle $\theta = 45^\circ$. The coefficient of kinetic friction between the block and the wedge is $\mu_k = 2/3$. The wedge is forced to accelerate horizontally to the left, and its acceleration varies with time as $a(t) = t \text{ m/s}^2$. At $t=0$, the block is released from rest relative to the wedge. Assuming the wedge is long enough so the block does not fall off, find the work done by the kinetic friction force on the block in the laboratory frame during the time interval $t=0$ to $t=2 \text{ s}$. (Take $g = 10 \text{ m/s}^2$)

A vertical, thermally insulated cylinder contains a rigid diatomic ideal gas. The cylinder is closed by a massless, thermally conducting piston of area $A$ that can move without friction. A block of mass $M$ rests on top of the piston. The external atmospheric pressure is $P_0$. The block is given an initial horizontal velocity $v_0$. Due to kinetic friction between the block and the piston, the block eventually comes to rest relative to the piston. The piston is constrained to move only vertically. Assuming all the heat generated by friction is absorbed by the gas, what is the ratio of the work done by the gas to the initial kinetic energy of the block?

A uniform conducting rod of length $L$ and resistance $r$ rotates with a constant angular velocity $\omega$ about one of its ends in a uniform magnetic field $B$ directed perpendicular to the plane of rotation. The free end of the rod slides on a fixed, frictionless, perfectly conducting circular ring. A stationary resistor of resistance $R$ is connected between the pivot and the ring. If the rod completes one full revolution, what is the work done by the magnetic force on the charge carriers, and the work done by the external mechanical agent to maintain the constant angular velocity, respectively?

A uniform solid sphere of mass $M$ and radius $R$ is restructured into a thin spherical shell of the same mass $M$ and radius $R$. The minimum energy required for this restructuring is $E_0$. A particle of mass $m$ is initially in a circular orbit of radius $2R$ around the solid sphere. The particle is given an instantaneous additional kinetic energy equal to $E_0$, and its velocity vector is simultaneously redirected to point radially inwards towards the center. It enters the newly formed shell through a small hole. Inside the shell, the particle experiences a constant non-conservative drag force of magnitude $f$. If the particle comes to rest exactly at the center of the shell, what is the value of $f$?

A damped mechanical oscillator of mass $m$, spring constant $k$, and damping coefficient $b$ is driven by a force $F(t) = F_0 \cos(\omega t)$. An analogous series LCR circuit with inductance $L=m$, capacitance $C=1/k$, and resistance $R=b$ is driven by a voltage $V(t) = F_0 \cos(\omega t)$. Both systems operate in steady state at their resonant frequency $\omega_0 = \sqrt{k/m}$. Let $E_{mech}$ be the total mechanical energy of the oscillator, and $U_{heat}$ be the heat dissipated in the LCR circuit over one complete cycle. Which of the following represents the ratio $\frac{E_{mech}}{U_{heat}}$?

A short electric dipole of constant dipole moment $\vec{p} = p_0 \hat{i}$ is placed in a region where the electrostatic field is given by $\vec{E} = E_0 \left( \frac{x^3}{a^3} - \frac{3x}{a} \right) \hat{i}$, where $E_0$ and $a$ are positive constants. The dipole is constrained to move only along the $x$-axis while maintaining its orientation. If the dipole is released from rest from its unstable equilibrium position and given an infinitesimal nudge towards the origin, what is the work done by the electric field on the dipole by the time it reaches its stable equilibrium position?

A cylindrical container filled with a liquid of density $\rho_f$ is accelerated vertically upwards with a time-varying acceleration $a(t) = \alpha t$, where $\alpha = 4 \text{ m/s}^3$. A small sphere of mass $m = 1 \text{ kg}$, volume $V$, and density $\rho_s = \rho_f / 2$ is released from rest at the bottom of the container at $t = 0$. The sphere experiences a relative velocity-dependent drag force $F_D = -b v$, where $b = 2 \text{ kg/s}$ and $v$ is the velocity of the sphere relative to the container. Assuming the container is sufficiently tall, what is the work done by the pseudo force on the sphere in the reference frame of the container during the interval $t = 0$ to $t = 1 \text{ s}$? (Take $g = 10 \text{ m/s}^2$)

A conducting rod of mass $m$ and length $l$ is placed on two long, parallel, smooth horizontal rails separated by a distance $l$. The rails are connected at one end by a resistor of resistance $R$ and an inductor of inductance $L$ in series. A uniform vertical magnetic field of magnetic induction $B$ exists in the region. At $t=0$, a constant horizontal force $F$ is applied to the rod, pulling it along the rails. The rod starts from rest. Let $W(t)$ be the total work done by the external force $F$ and $H(t)$ be the total heat dissipated in the resistor from time $0$ to $t$. Find the exact value of $\lim_{t \to \infty} [W(t) - H(t)]$.