Gravitation - Class 11 Part 1

A fixed solid non-conducting sphere of radius $R$ has a non-uniform mass density $\rho_m = \rho_0 r$ and a uniform volume charge density $\rho_e$, where $r$ is the radial distance from the center. A point particle of mass $m$ and charge $-q$ ($q > 0$) is released from rest at the surface of the sphere. Assuming the particle can move freely through the sphere without any resistance, which of the following expressions represents the speed of the particle when it reaches the center of the sphere? (Given $G$ is the universal gravitational constant and $\epsilon_0$ is the permittivity of free space).

Two fixed point masses $M$ and $4M$ are separated by a distance $L$ in an inertial reference frame. A spaceship is moving along the straight line joining the two masses. At a particular instant, the spaceship has an acceleration of magnitude $a = \frac{12GM}{L^2}$ directed towards the mass $4M$. Inside the spaceship, a small test particle is released. At what instantaneous distance from the mass $M$ (along the line segment joining the masses) should the particle be released so that its initial acceleration relative to the spaceship is zero?

Two heavy point masses, each of mass $M$, are fixed at coordinates $(d, 0, d)$ and $(-d, 0, d)$ in a standard $xyz$ coordinate system where the z-axis is vertical. A small block of mass $m$ moves along the y-axis on a rough horizontal table lying in the $xy$-plane ($z=0$). The coefficient of kinetic friction between the block and the table is $\mu$. The block is moved from $A(0, d, 0)$ to the origin $O(0, 0, 0)$. Earth's uniform gravitational field $g$ acts in the $-z$ direction. Assuming the block never loses contact with the table, what is the sum of the work done on the block by the gravitational field of the two masses and the force of friction?

Two concentric thin spherical shells of masses $M$ and $2M$ have radii $R$ and $2R$ respectively. What is the total gravitational potential energy stored in the gravitational field strictly in the region $r \geq 2R$?

Two concentric thin conducting spherical shells have radii $R$ and $3R$ (where $R$ is in meters). The inner shell is connected to the ground via a thin conducting wire, while the outer shell is isolated and given a total positive charge $+Q$ (in Coulombs). Let $E_1$ be the magnitude of the electric field just outside the surface of the inner shell, and $E_2$ be the magnitude of the electric field just outside the surface of the outer shell. Determine the dimensionless ratio $\frac{E_1}{E_2}$.

A massive spherical cloud of dust of radius $R$ has a non-uniform mass density $\rho(r) = \rho_0 \frac{R}{r}$ for $r \le R$, where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. A particle of mass $m$ is initially moving in a circular orbit of radius $r_0$ ($r_0 < R$) inside the cloud. The particle is given a small radial impulse, causing it to undergo small radial oscillations. What is the angle $\Delta \Theta$ swept by the particle's radius vector between two successive points of minimum radial distance?

A planet of radius $R$ has a spherically symmetric, non-uniform mass density profile given by $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$, where $r$ is the distance from the center. A straight, smooth tunnel is excavated through the planet, passing at a perpendicular distance $d = \frac{R}{2}$ from the center. A particle of mass $m$ is placed in the tunnel. It experiences a viscous drag force $F_{\text{drag}} = -b v$ (where $v$ is its velocity) and is driven by an external periodic force $F_{\text{ext}} = F_0 \cos(\omega t)$ directed along the tunnel. Assuming small oscillations about the midpoint of the tunnel ($x \ll R$), what is the resonant frequency $\omega_r$ at which the steady-state amplitude of the particle's oscillation is maximized? (Assume the system is underdamped).

An inner spherical shell of radius $R$ has a surface mass density $\sigma_1(\theta) = \sigma_0 (1 + 3\cos^2 \theta)$. An outer concentric spherical shell of radius $2R$ has a surface mass density $\sigma_2(\theta) = \sigma_0 (1 - \cos^2 \theta)$, where $\theta$ is the polar angle. A particle of mass $m$ is placed at a distance $r = \frac{3}{2}R$ from the center on the polar axis ($\theta = 0$). Find the magnitude of the net gravitational force on the particle.

A rocket of mass $m$ is propelled radially outward from the center of a uniform solid spherical planet of mass $M$ and radius $R$. The rocket maintains a constant speed $v$ relative to the planet throughout its journey to infinity. The rocket's engine provides thrust solely to overcome the planet's gravitational pull. Let $P(t)$ be the instantaneous mechanical power delivered by the engine. Determine the discontinuity in the rate of change of power, defined as $\Delta \dot{P} = \lim_{\epsilon \to 0^+} \left( \left. \frac{dP}{dt} \right|_{r=R+\epsilon} - \left. \frac{dP}{dt} \right|_{r=R-\epsilon} \right)$.

A solid sphere of mass $M_0$ and radius $R$ is fixed in space. A thick spherical shell of inner radius $R$ and outer radius $2R$, having a total mass $M_s$, is slowly assembled around the sphere by bringing particles from infinity. The density of the assembled shell varies with radial distance $r$ from the center as $\rho(r) = \frac{c}{r}$, where $c$ is a positive constant. What is the work done by the external agent in assembling the shell?

A uniform solid sphere of initial mass $M$ (in $\text{kg}$) and radius $R$ (in $\text{m}$) has a spherical cavity of radius $R/2$ created within it. The center of the cavity is located at a distance $R/2$ from the center of the original solid sphere. Let $V_1$ and $V_2$ (in $\text{J}/\text{kg}$) be the gravitational potentials at the center of the original sphere and the center of the cavity, respectively. If $G$ is the universal gravitational constant, determine the potential difference $V_2 - V_1$.

A space probe of initial mass $m_0$ is released from rest at a distance $R_0$ from the center of a planet of mass $M$. As it falls radially inwards under the planet's gravity, it sweeps up a stationary cloud of cosmic dust. The mass of the probe increases such that its mass at a distance $r$ from the center is given by $m(r) = m_0 \frac{R_0}{r}$. Assuming the dust particles are initially at rest with respect to the planet and the accretion is completely inelastic, what is the speed of the probe when it reaches a distance $R$ ($R < R_0$) from the center? (Ignore any atmospheric drag or forces other than gravity).

Two identical uniform solid disks, each of mass $M$ and radius $R$, are placed parallel to the $xy$-plane such that their centers lie on the $z$-axis. The first disk is centered at the origin $(0,0,0)$ and the second disk is centered at $(0,0,3R)$. A student is tasked with finding the absolute gravitational potential at the point $P(0,0,4R)$ on the $z$-axis, with the standard reference $V \to 0$ as $r \to \infty$. Which of the following represents the correct potential at $P$?

Two isolated stars of masses $m$ and $2m$ interact only via their mutual gravitational force. In the center of mass reference frame, the system has a constant total angular momentum $L$. The relative radial motion of the stars can be modeled as a single particle of equivalent mass moving in a 1D effective potential $V_{eff}(r)$, where $r$ is the distance between the stars. What is the radius $r_c$ of the stable circular orbit and the corresponding minimum effective potential $V_{min}$?

A thin non-conducting rod of length $2L$ lies along the $x$-axis, centered at the origin. It carries a non-uniform linear charge density $\lambda(x) = \lambda_0 \frac{|x|}{L}$, where $\lambda_0$ is a positive constant. An external agent moves a point charge $q$ from point $A(0, L, 0)$ to point $B\left(0, 0, \frac{3}{4}L\right)$ along a complex curved path $\Gamma$ parameterized by $\vec{r}(t) = L\sin(\pi t)\hat{i} + L\cos^2\left(\frac{\pi t}{2}\right)\hat{j} + \frac{3}{4}L\sin^2\left(\frac{\pi t}{2}\right)\hat{k}$ for $t \in [0,1]$. What is the work done by the electric field on the charge $q$ during this motion?

A particle of mass $m$ and charge $q$ is located at the origin of a coordinate system. At time $t=0$, it is projected with an initial velocity $\vec{v} = v_0 \hat{i}$. A time-dependent, spatially uniform electric field $\vec{E} = c t \hat{i}$ is present in the region, where $c$ is a positive constant. What is the work done by the electric field on the particle during the time interval from $t=0$ to $t=T$?

A particle of mass $m$ is projected from infinity with an initial speed $v_0$ and impact parameter $b$ towards a fixed force center. The potential energy of the interaction is given by $V(r) = -\frac{k}{r^3}$, where $k > 0$ is a constant and $r$ is the radial distance from the center. To avoid spiraling into the origin, the impact parameter $b$ must be greater than or equal to a critical value $b_c$. Find $b_c$.

A planet of mass $M$ and cross-sectional area $A_p$ is in a nearly circular orbit of radius $R$ around a star of mass $M_s$. The star emits a steady, spherically symmetric stellar wind of gas radially outwards at a constant speed $v_w$. The total mass loss rate of the star due to this wind is $\dot{m}$. The planet experiences a tangential drag force of magnitude $F_D = C_d \rho A_p v^2$, where $\rho$ is the local gas density, $v$ is the planet's orbital speed, and $C_d$ is the drag coefficient. Assuming $M \ll M_s$ and $v_w \ll v$, what is the magnitude of the rate of change of the planet's areal velocity, $\left| \frac{dV_A}{dt} \right|$?

A satellite of mass $m$ and constant magnetic dipole moment $\vec{M} = M_0 \hat{i}$ orbits a planet. The planet's magnetic field in the orbital plane ($xy$-plane) is given by $\vec{B} = B_0 \left( \frac{x^2 - y^2}{R^2} \hat{i} - \frac{2xy}{R^2} \hat{j} \right)$, where $R$ is a positive constant. At the instant the satellite passes through the point $P \left( \frac{R}{\sqrt{2}}, \frac{R}{\sqrt{2}} \right)$, what is the magnitude of the rate of change of its areal velocity vector about the origin?

A comet of mass $m$ is in a parabolic orbit around a massive star of mass $M$ ($M \gg m$). At its perihelion distance $r_0$, it passes through a localized, stationary cloud of cosmic dust. The comet sweeps up a mass $\Delta m = \frac{m}{3}$ from the cloud in a completely inelastic collision. The energy dissipated during this process is entirely absorbed by the comet, exciting its internal vibrational modes. Which of the following gives the energy $E_{vib}$ absorbed by the vibrational modes and the new semi-major axis $a'$ of the comet's orbit?

A planet of initial mass $m_0$ is in a circular orbit of radius $r_0$ and period $T_0$ around a star of mass $M$ ($M \gg m_0$). It moves through a uniform cosmic dust cloud, accreting dust at a constant rate $\frac{dm}{dt} = \alpha$. The dust is initially at rest relative to the star. Additionally, the cloud exerts a viscous drag force on the planet given by $\vec{F}_d = -\beta \vec{v}$, where $\vec{v}$ is the planet's velocity and $\beta$ is a positive constant. Assuming the orbit remains quasi-circular, what is the orbital period $T$ of the planet when its mass reaches $2m_0$?

A spherically symmetric planet of radius $R$ has an internal density profile $\rho(r) = \rho_0 \left( 1 - \frac{r}{R} \right)^2$ for $r \le R$. A satellite orbits the planet in a circular trajectory just above its surface with an orbital period $T_0$. Due to internal thermal processes, the planet undergoes a structural phase transition, expanding to a new radius $R'$ while strictly conserving its total mass. Its new spherically symmetric density profile becomes $\rho'(r) = \frac{\rho_0}{32} \left( 1 - \frac{r^2}{R'^2} \right)$ for $r \le R'$. If the satellite's orbit is adjusted so that it again orbits just above the new surface $R'$, what is the new orbital period $T'$?

A satellite of mass $m$ is in a quasi-circular orbit of initial radius $r_0$ and initial velocity $v_0$ around a planet. A uniform magnetic field perpendicular to the orbital plane exists in the region, varying with time as $B(t) = \frac{B_0}{\sqrt{1 + \alpha t}}$, where $B_0$ and $\alpha$ are positive constants. The satellite deploys a straight conducting tether of length $L$ ($L \ll r_0$) oriented radially outward. The tether forms a local closed electrical circuit of constant total resistance $R$ with the surrounding plasma. Assuming the circuit's enclosed area is negligible (so the induced EMF is purely motional) and neglecting all other forms of drag, which of the following expressions gives the orbital velocity $v(t)$ of the satellite as a function of time?

A small uncharged satellite orbits a spherical planet of rest mass $M$ and radius $R$ in a circular orbit. The initial orbital period is $T_0$. The planet is then given a total charge $Q$ distributed uniformly throughout its volume. According to mass-energy equivalence, the electrostatic self-energy of the planet contributes to its effective gravitational mass. If $c$ is the speed of light in vacuum, $\epsilon_0$ is the vacuum permittivity, and the added mass equivalent is much smaller than $M$, which of the following expressions best represents the new orbital period $T$ of the satellite?

A hypothetical spherical planet of radius $R$ is formed by assembling dust particles from infinity. The 'bare' density profile (defined as the sum of the rest masses of the constituent dust particles per unit volume) is given by $\rho(r) = \frac{k}{r}$ for $0 < r \le R$, where $k$ is a positive constant. Due to mass-energy equivalence, the gravitational binding energy released during formation reduces the planet's effective gravitational mass. Assuming the binding energy is calculated using the bare mass distribution, what is the escape velocity for a small test mass launched from the surface of this planet? ($G$ is the universal gravitational constant, $c$ is the speed of light in vacuum)

A particle of mass $m$ is projected radially outwards from the surface of a planet of mass $M$ and radius $R$. The planet is surrounded by an atmosphere that exerts a drag force on the particle given by $F_d = -\frac{\alpha m}{r^2} v^2$, where $v$ is the instantaneous velocity of the particle, $r$ is its distance from the center of the planet, and $\alpha$ is a positive constant with dimensions of length. What is the minimum initial velocity $v_0$ required for the particle to escape the planet's gravitational pull?

A uniform solid sphere of mass $M$ and radius $R$ carries a total charge $Q$ uniformly distributed throughout its volume. The gravitational potential $V_g$ produced by the sphere is defined to be zero at infinity. The electrostatic potential $V_e$ produced by the sphere is defined to be zero at its surface. A point particle of mass $m$ and charge $q$ is placed at the center of the sphere. What is the potential energy of the point particle in the fields of the sphere?

A solid sphere of radius $R$ and total mass $M$ has a radially varying mass density $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$, where $r$ is the distance from the center. A narrow, smooth radial tunnel is drilled into the sphere. A small particle of mass $m$ is projected from a depth $r = \frac{R}{2}$ inside the tunnel towards the surface. What is the minimum projection speed $v_0$ required for the particle to escape the gravitational pull of the sphere?

A spherical fluid body of mass $M$ and radius $R$ is initially at rest at a height $h$ in a uniform external gravitational field $g$. As it falls freely through the distance $h$, it splits into $8$ identical spherical droplets that move far apart from each other. The fluid is incompressible. Assuming the surface tension energy is negligible compared to the gravitational self-energy, and no energy is dissipated as heat, what is the total kinetic energy of the system of droplets after falling the distance $h$?

Two point particles of rest masses $m_1$ and $m_2$ are initially held at rest at a distance $R$ from each other in an inertial frame in deep space. They are released and fall towards each other under mutual gravitational attraction. Upon collision, they completely fuse into a single composite particle of rest mass $M$, and a single photon is emitted in the process. Assuming no other energy losses and that the system is isolated, what is the energy $E$ of the emitted photon? (Here $c$ is the speed of light in vacuum, and $G$ is the gravitational constant).

A satellite of initial rest mass $m_0$ is in a stable circular orbit of radius $r_0$ around a planet of mass $M$ ($M \gg m_0$). The satellite carries a nuclear power source. Over a period of time, the reactor converts a mass defect $\Delta m$ into energy, which is radiated away isotropically in the satellite's rest frame. Assuming no other forces act on the system and ignoring relativistic velocity effects ($v \ll c$), what is the final orbital radius $r_f$ and the final angular momentum $L_f$ of the satellite about the center of the planet?

A smooth circular wire of radius $r$ is fixed concentrically around a solid non-conducting sphere of mass $M$ and uniform charge $+Q$. A small bead of mass $m$ and charge $-q$ (where $Q, q > 0$) slides along the wire in uniform circular motion. The system is in deep space, isolated from other external fields. If the bead experiences a state of 'weightlessness' (i.e., the normal force exerted by the wire on the bead is exactly zero), which of the following expressions correctly gives the orbital speed $v$ of the bead?

A satellite of mass $M_s$ is in a circular orbit of radius $R_0$ around a planet of mass $M$. The satellite is tidally locked, maintaining a constant orientation relative to the planet. Inside the satellite, a smooth straight tube is aligned along the radial direction. Two pistons, each of mass $m$ ($m \ll M_s$), are placed in the tube symmetrically on opposite sides of the satellite's center of mass. Between the pistons is $n$ moles of an ideal gas maintained at a constant temperature $T$. An internal mechanical system exerts forces on the pistons to ensure they move quasi-statically from a distance $L$ to $2L$ from the center of mass. What is the total work done by this mechanical system on the pistons? (Let $R$ be the ideal gas constant)

Two identical massive spheres, each of mass $M$, are fixed at positions $(d, 0, 0)$ and $(-d, 0, 0)$ in free space. A small rigid probe of length $L$ ($L \ll d$) moves along a circular equipotential path of radius $R=d$ in the $y-z$ plane, centered at the origin. The probe maintains its orientation parallel to the $x$-axis throughout its motion. Although the probe's center of mass is in free fall, a tidal force acts across it. What is the magnitude of the relative tidal acceleration between the two ends of the probe?

A uniform spherical planet of mass $M$ and initial radius $R$ rotates about its axis with an initial angular velocity $\omega$. The coefficient of linear thermal expansion of the planet's material is $\alpha$. The planet undergoes a uniform temperature increase of $\Delta T$. Assuming no external torques act on the planet and $\alpha \Delta T \ll 1$, what is the approximate change in the effective acceleration due to gravity ($\Delta g_{eff}$) at the planet's equator? Let $g = \frac{GM}{R^2}$ be the initial true acceleration due to gravity at the surface.

A motorized gyroscope consists of a rotor of mass $M$ and moment of inertia $I$ about its spin axis. It is mounted on a light, rigid horizontal axle of total length $2l$, with the rotor at the exact center. The ends of the axle rest on two identical digital weighing scales, $S_1$ and $S_2$. The rotor spins at a constant angular speed $\omega$ such that its angular momentum vector $\vec{L}$ points exactly towards $S_1$. The setup is located at a latitude $\lambda$ in the Northern Hemisphere, and the Earth rotates with a constant angular speed $\Omega$. The axle is oriented at an angle $\theta$ North of East. Horizontal guides keep the axle stationary relative to the Earth without exerting any vertical forces. If $N_1$ and $N_2$ are the vertical normal forces exerted by scales $S_1$ and $S_2$ respectively, what is the value of $N_2 - N_1$?

A spherical planet of radius $R$ rotates about its axis with a constant angular velocity $\omega$. The density of the planet varies with the radial distance $r$ from its center as $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$, where $\rho_0$ is a positive constant. What is the effective acceleration due to gravity (radial component) measured by an observer located inside the planet at a radial distance $r = \frac{R}{2}$ and at a latitude $\lambda = 60^\circ$?

A perfectly spherical planet of mass $M$ and radius $R$ rotates with a constant angular velocity $\omega$. A particle of mass $m$ is initially at rest at the North Pole of the planet. An external agent slowly transports the particle along the planet's surface to the equator, and then lifts it radially to a synchronous orbit of radius $R_s$ (where the particle's orbital angular velocity matches the planet's rotation). In the rotating frame of the planet, what is the total work done by the external agent? (Assume no other celestial bodies exert gravitational influence).

A rocket of initial mass $M_0$ is launched vertically upwards from the Earth's equator. It ejects gas vertically downwards (relative to the rocket) at a constant relative speed $u$ and a constant rate $\mu = \frac{M_0}{\tau}$. The Earth rotates with a constant angular speed $\omega$. Neglecting air resistance, the variation of gravity with height, and assuming the lateral drift is small enough that the rocket's orientation remains perfectly vertical relative to the Earth, which of the following expressions gives the magnitude of the lateral drift velocity of the rocket at time $t = \frac{\tau}{2}$? (Let $g$ be the true acceleration due to gravity and $R$ be the radius of the Earth).

A satellite of instantaneous mass $m$ is in a nearly circular orbit around a planet of mass $M$ ($M \gg m$). The satellite passes through a uniform, stationary dust cloud, accreting dust at a constant rate $\frac{dm}{dt} = \lambda$. Due to this accretion, the satellite's instantaneous equilibrium circular orbit radius $r_c$ slowly shrinks. The satellite undergoes small radial oscillations $x$ such that its radial distance is $r(t) = r_c + x(t)$, where $x \ll r_c$. Assume the fractional change in mass during one oscillation is negligible. The differential equation governing these small radial oscillations can be mapped to the equation of a damped L-C-R circuit: $L_e \frac{d^2q}{dt^2} + R_e \frac{dq}{dt} + \frac{q}{C_e} = 0$. If the radial displacement $x$ is analogous to the charge $q$, and the instantaneous mass $m$ is analogous to the inductance $L_e$, what are the equivalent resistance $R_e$ and capacitance $C_e$ of the circuit?

A spherically symmetric planet of radius $R$ has a non-uniform mass density given by $\rho(r) = \frac{\rho_0 R}{r}$ for $r \le R$, where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. A satellite of mass $m$ is in a circular orbit around this planet at a radial distance of $r = 2R$. A particle, also of mass $m$, is projected from the center of the planet ($r=0$) through a narrow, smooth radial tunnel. If the particle's speed when it reaches $r = 2R$ is exactly equal to the orbital speed of the satellite, what is the required projection speed $v_0$ at the center? (Assume no other celestial bodies are present and $G$ is the universal gravitational constant).

A satellite of mass $m$ is in a circular orbit of radius $R$ in the $xy$-plane around the Earth (mass $M$, fixed at the origin of an inertial frame). The satellite's orbital angular velocity is $\omega = \sqrt{GM/R^3}$. At time $t=0$, the satellite crosses the positive $x$-axis.

A non-inertial reference frame $S'$ translates relative to the Earth. The origin of $S'$ coincides with the Earth's center at $t=0$ and moves along the positive $x$-axis with a constant acceleration $\vec{a}_s = a_0 \hat{i}$ and zero initial velocity.

Calculate the work done strictly by the Earth's gravitational force on the satellite, as measured by an observer in frame $S'$, during the time interval from $t=0$ to $t=\pi/\omega$.

A uniform solid spherical satellite of mass $m$ and radius $a$ ($a \ll r_0$) is initially in a circular geostationary orbit of radius $r_0$ around the Earth (mass $M$). Initially, the satellite is irrotational relative to the distant stars. An internal motor, powered by onboard batteries, spins the satellite up until it achieves a 'pure rolling' state along its orbit. In this state, its spin angular velocity $\omega$ and its new orbital speed $v$ satisfy $v = \omega a$, with the spin and orbital angular momentum vectors being parallel. Assuming no mass is ejected and no external torques act on the satellite about the Earth's center, find the approximate change in the orbital radius $\Delta r = r - r_0$.

A satellite is in a circular orbit of radius $R$ around a planet of mass $M$. As part of its structure, it contains a uniform straight rod of mass $m$, length $L$ ($L \ll R$), cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$. The rod is gravity-gradient stabilized such that it always aligns along the radial direction towards the center of the planet. The rod is mounted inside a rigid, massless, non-expanding frame that holds its ends, strictly constraining its total length to remain $L$. If the temperature of the rod increases by $\Delta T$, what is the magnitude of the compressive force $F_c$ exerted by the frame on each end of the rod? (Assume the orbital radius $R$ remains constant and $G$ is the universal gravitational constant).

A geostationary satellite is in a circular orbit of radius $R$ around a planet. The planet and the satellite both have an angular velocity $\omega$. A maintenance rocket initially docked with the satellite detaches and instantaneously acquires a radial velocity $v_0 = \frac{\omega R}{3}$ relative to the satellite, directed away from the planet's center. Immediately after detachment, what is the radius of curvature of the rocket's trajectory as observed in a reference frame rotating with angular velocity $\omega$ (the frame in which the satellite is at rest)?

A massive space station is in a circular orbit of radius $r_0$ (in $\text{m}$) around a planet of mass $M$ (in $\text{kg}$). The station is tidally locked, rotating with a constant orbital angular velocity $\omega_0 = \sqrt{\frac{GM}{r_0^3}}$ (in $\text{rad/s}$) to keep the same face towards the planet. Inside the station, a smooth straight tube is aligned along the radial direction. A payload of mass $m$ (in $\text{kg}$) is placed inside the tube at the equilibrium position $r = r_0$. It is connected to the station by a spring of force constant $k$ (in $\text{N/m}$) and a damper providing a resistive force $F_d = -b v_{rel}$, where $v_{rel}$ (in $\text{m/s}$) is the radial velocity of the payload relative to the tube and $b$ (in $\text{kg/s}$) is the damping constant.

An engineer maps the small radial displacements $x$ of the payload to an equivalent electrical $L$-$C$-$R$ circuit equation $L_e \frac{d^2q}{dt^2} + R_e \frac{dq}{dt} + \frac{1}{C_e} q = 0$. In this analogy, the displacement $x$ maps to charge $q$, and the mass $m$ maps to inductance $L_e$. Assuming $k > 3m\omega_0^2$, what is the expression for the equivalent capacitance $C_e$?

A satellite of mass $m$ is in a circular orbit of radius $r$ around the Earth (mass $M$). The thin atmosphere exerts a tangential drag force of magnitude $F_d = \alpha v^2$ on the satellite, where $v$ is its orbital speed and $\alpha$ is a positive constant.

In Scenario A, the satellite's onboard engine provides a continuous tangential thrust to maintain the satellite in the exact same circular orbit of radius $r$. Let $P_A$ be the mechanical power delivered by the engine.

In Scenario B, the engine is turned off, and the satellite's orbit slowly decays while remaining approximately circular. Let $P_B$ be the rate at which the gravitational potential energy of the satellite-Earth system decreases at the instant the orbital radius is $r$.

What is the ratio $\frac{P_A}{P_B}$?

A uniform solid sphere of mass $M$ and initial radius $R_0$ is isolated in deep space. The thermal internal energy of the material is given by $U_{\text{th}} = M c T$, where $c$ is a constant and $T$ is the absolute temperature. An internal heat source slowly raises the uniform temperature of the sphere by an amount $\Delta T$. Due to thermal expansion, the radius of the sphere increases to $R = R_0(1 + \alpha \Delta T)$, where $\alpha$ is a constant. Assuming the sphere's mass density remains uniform throughout its volume during the expansion, what is the total heat $Q$ supplied to the sphere? ($G$ is the universal gravitational constant)

A solid planet of radius $R \text{ (in m)}$ and total mass $M \text{ (in kg)}$ has a non-uniform, spherically symmetric density distribution $\rho(r) = \rho_0 \left(1 - \frac{r^2}{R^2}\right) \text{ (in kg m}^{-3}\text{)}$, where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. A narrow, frictionless tunnel is drilled along a diameter of the planet. A particle of mass $m \text{ (in kg)}$ is placed inside the tunnel at a distance $x_0 \text{ (in m)}$ from the center, such that $x_0 \ll R$, and released from rest. If $g \text{ (in m s}^{-2}\text{)}$ is the acceleration due to gravity at the surface of the planet, what is the time period $T \text{ (in s)}$ of the resulting small oscillations of the particle?

A solid spherical planet of mass $M$ and radius $R$ initially has a uniform mass density. It carries a total charge $Q$ uniformly distributed over its surface. A concentric thin spherical shell of mass $m$ and radius $2R$ carries a total charge $-Q$ uniformly distributed over its surface. The shell slowly collapses under the combined action of gravity and electrostatic forces until it rests on the surface of the planet. During this process, the planet undergoes a structural phase transition, changing its mass density profile from uniform to $\rho(r) = \rho_0 (1 - r/R)$ while maintaining its total mass $M$, radius $R$, and surface charge $Q$. What is the total work done by the conservative forces (gravitational and electrostatic) of the system during this entire process?