Laws of Motion - Class 11 Part 1

A rigid diathermic container of negligible mass and internal volume $V$ rests on a frictionless horizontal surface. It initially contains an ideal gas of mass $m_0$ and molar mass $\mu$ at temperature $T$. A small aperture of area $A$ is opened on one side, allowing the gas to effuse into a vacuum. The gas temperature is maintained constant at $T$ by the surroundings. The effusing gas exerts a recoil thrust $F = PA$, where $P$ is the instantaneous internal pressure. Using the kinetic theory of gases for the effusion rate, determine the displacement of the container by the time the mass of the gas inside reduces to $\frac{m_0}{2}$.

A block of initial mass $M_0 = 10 \text{ kg}$ rests on a rough inclined plane angled at $\theta = 37^\circ$ to the horizontal. The coefficient of static friction is $\mu_s = 0.5$. At $t = 0$, the block begins to eject gas at a constant rate $\lambda = 1 \text{ kg/s}$ with a relative exhaust velocity $v_{\text{rel}} = 50 \text{ m/s}$. The gas is ejected such that the resulting thrust force on the block is directed up the incline at an angle $\alpha = 37^\circ$ away from the incline's surface. Assuming $g = 10 \text{ m/s}^2$, at what time $t$ does the block first begin to slide? (Take $\sin 37^\circ = 3/5$, $\cos 37^\circ = 4/5$)

A smooth wedge of mass $M$ and angle $\theta$ rests on a smooth horizontal surface. The inclined surface faces right, meaning its height decreases towards the right. A small block of mass $m$ is placed on the incline. A horizontal force $F(t) = mg \sin\theta \cos\theta + c t$ (where $c$ is a positive constant and $t$ is time) is applied to the wedge towards the left. If the system starts from rest at $t=0$, which of the following expressions gives the relative velocity of the block with respect to the wedge down the incline as a function of time $t$?

A block of mass $m$ is released from rest on a wedge of mass $M$ and inclination angle $\theta$. The wedge rests on a smooth horizontal surface. The coefficient of kinetic friction between the block and the wedge varies as $\mu = kx$, where $x$ is the distance the block has slid relative to the wedge down the incline. Which of the following expressions correctly represents the normal force $N$ exerted by the wedge on the block as a function of $x$?

A fixed solid non-conducting sphere of radius $R$ with a uniform volume charge density $\rho > 0$ is located beneath a fixed, smooth, non-conducting inclined plane of angle $\theta$. The inclined plane is tangent to the sphere at a point $P$. A small block of mass $m$ and charge $q > 0$ is released from rest on the incline at a distance $\sqrt{3}R$ along the incline from $P$, higher up the plane. Given that the block just loses contact with the plane as it reaches $P$, what is the speed of the block at $P$? (Assume acceleration due to gravity is $g$ and permittivity of free space is $\epsilon_0$)

A toy rocket of initial mass $m_0$ kg is placed on the smooth horizontal floor of a train. The train accelerates forward with a constant acceleration $a_0 \text{ m/s}^2$. The rocket is pointed forward. At $t=0$ s, the rocket motor is ignited, ejecting gas backwards at a constant mass rate $\mu \text{ kg/s}$ and at a constant speed $u \text{ m/s}$ relative to the rocket. Given that the constant thrust provided by the motor is $u\mu = \frac{1}{2} m_0 a_0$, what is the maximum backward speed of the rocket relative to the train in m/s? (Assume the rocket has sufficient fuel and does not hit the back wall of the train).

A perfectly conducting rod of length $L$ rotates with a constant angular velocity $\omega$ in a horizontal plane about a fixed vertical axis passing through one of its ends. A uniform vertical magnetic field $B$ exists in the region, directed downwards. The rotation of the rod is counterclockwise when viewed from above. If $m_e$ and $-e$ are the mass and charge of an electron respectively, what is the exact potential difference $V_{\text{center}} - V_{\text{end}}$ between the pivoted end (center) and the free end of the rod in the steady state?

A wedge of mass $M = 3m$ with an inclination angle $\theta = 30^\circ$ rests on a smooth horizontal surface. The inclined surface faces left (elevation increases to the right). A thin hollow spherical shell of mass $m$ and radius $R$ is placed on the incline. A horizontal force $F$ is applied to the wedge to the left. Assuming the shell rolls without slipping on the wedge, what is the acceleration of the wedge $a_w$?

A particle of mass $m$ and charge $q$ is projected with an initial speed $v_0$ in the $xy$-plane. A time-varying uniform magnetic field $\vec{B} = B_0(1 + \alpha t)\hat{k}$ exists in the region, where $B_0$ and $\alpha$ are positive constants. The particle experiences a velocity-dependent drag force $\vec{F}_d = -k \vec{v}$, where $k$ is a positive constant. Assuming the effects of the induced electric field are negligible, what is the magnitude of the total angle rotated by the velocity vector of the particle from $t = 0$ until the moment its speed reduces to $v_0/e$?

A pendulum consists of a uniform solid sphere of mass $M$ and radius $R = L/2$ attached to a massless rigid rod of length $L/2$. The rod's other end is pivoted at a fixed point $O$, making the distance from $O$ to the sphere's center of mass equal to $L$. The pendulum is initially at rest in the vertical position. A horizontal impulse $J$ is applied to the lowest point of the sphere. Assuming the maximum angular deflection $\theta_0$ is small, which of the following expressions gives $\theta_0$?

A small electric dipole of dipole moment $p$ and moment of inertia $I$ is placed in a non-uniform electric field $\vec{E} = E_0 \left( \frac{x^4}{4L^4} - \frac{x^2}{2L^2} \right) \hat{i}$, where $E_0$ and $L$ are positive constants. The dipole is free to move along the $x$-axis and rotate in the $xy$-plane. If the dipole is slightly rotated from its fully stable equilibrium configuration, what is the angular frequency of its small rotational oscillations?

A wedge with an angle of inclination $\theta$ is initially moving horizontally to the left with a constant acceleration $a$. A block of mass $m$ is placed on the smooth inclined surface of the wedge and is attached to a light spring of spring constant $k$. The other end of the spring is fixed to the top of the wedge. The block is initially at rest relative to the wedge. Suddenly, the acceleration of the wedge is reversed, maintaining the same magnitude $a$ but now directed to the right. Assuming the block always remains in contact with the wedge, what is the amplitude of the subsequent simple harmonic motion of the block relative to the wedge?

A block is placed on a smooth horizontal floor and is attached to a relaxed horizontal ideal spring of stiffness $k$. The other end of the spring is fixed to a vertical wall. A light inextensible string is tied to the block, passes over a small smooth fixed pulley located at a height $h$ above the floor, and hangs vertically downwards. The lower end of the string is attached to the free bottom end of a uniform vertical metallic rod of length $L$, cross-sectional area $A$, and Young's modulus $Y$. The upper end of the rod is rigidly fixed to a horizontal ceiling. Initially, the block is at a horizontal distance $x$ from the vertical line passing through the pulley, the string is just taut, and the rod is at temperature $T_0$. The temperature of the rod is slowly increased by $\Delta T$. Assuming the block's displacement is small, what is the work done by the string on the block?

A block of mass $M = 18.75 \text{ kg}$ rests on a smooth horizontal table. A long, light string of linear mass density $\mu = 0.01 \text{ kg/m}$ is attached to the block, passes around a light movable pulley, and has its other end fixed to a vertical wall. The string segments are horizontal. At $t=0$, the movable pulley is pulled away from the wall with a time-varying acceleration $a_p = 6t \text{ m/s}^2$. At the exact same instant $t=0$, a transverse wave pulse is generated at the block. The initial distance between the block and the pulley is $L_0 = 101 \text{ m}$. Assuming the tension is uniform along the string at any given instant, find the time $t_0$ when the pulse reaches the pulley.

Two particles $A$ and $B$ of masses $m$ and $2m$ respectively lie on a smooth horizontal $xy$-plane. They are connected by a light inextensible string of length $L$. Initially, particle $A$ is at rest at the origin $(0,0)$ and particle $B$ is at rest at $(\frac{3L}{5}, 0)$. Particle $B$ is suddenly projected with a velocity $v_0 \hat{j}$ along the positive $y$-direction while $A$ remains free to move. What is the magnitude of the velocity of particle $A$ just after the string becomes taut?

A small square loop of mass $m$, side length $a$, and electrical resistance $R$ is constrained to move along the positive $x$-axis on a smooth horizontal surface. A light inextensible string is attached to the loop, passes over a fixed massless pulley at $x = D$, then goes around a movable massless pulley, and is finally attached to a fixed peg at $x = D$. All string segments are parallel to the $x$-axis. A block of mass $M$ is attached to the movable pulley and slides on the same horizontal surface. A non-uniform magnetic field $\vec{B} = B_0 x^2 \hat{k}$ exists in the region. At $t=0$, the loop is at $x=0$ and is given an initial velocity $V_0$ in the $+x$ direction. Find the maximum $x$-coordinate reached by the loop before it momentarily comes to rest. (Assume $a \ll x$ and $D$ is sufficiently large)

A wedge with an angle of inclination $\theta = 45^\circ$ rests on a horizontal surface. The inclined surface of the wedge faces upwards and to the right. The wedge is given a constant horizontal acceleration $a = 10 \text{ m/s}^2$ to the left. A block of mass $m$ is placed on the incline and projected up the incline with an initial velocity $v_0 = 10 \text{ m/s}$ relative to the wedge. If the coefficient of kinetic friction between the block and the wedge is $\mu_k = 0.5$, find the maximum distance $d$ the block travels up the incline relative to the wedge. (Take $g = 10 \text{ m/s}^2$)

A solid cylinder of mass $M$ and radius $R$ is free to rotate about its fixed horizontal axis. Initially, it rotates with angular velocity $\omega_0$. A small block of mass $m$ is placed at the top of the cylinder ($\theta = 0$). An external mechanism forces the block to slide down the surface of the cylinder, in the direction of its rotation, at a constant speed $v_0 = \sqrt{gR/2}$ relative to the ground. The mechanism only exerts a tangential force on the block. The coefficient of kinetic friction between the block and the cylinder is $\mu$. Assume the cylinder's surface always moves faster than the block ($\omega R > v_0$). Find the angular velocity of the cylinder when the block reaches an angular position $\theta = \pi/6$ from the top.

A small block of mass $m$ is placed on a rough articulating wedge. The wedge's angle of inclination $\theta$ with the horizontal is dynamically adjusted over time $t$ such that $\theta(t) = \frac{\pi}{2T^2} t^2$, where $T$ is a given positive time constant. The block is kept stationary relative to the wedge by a string parallel to the incline. Which of the following represents the time-averaged normal force exerted by the wedge on the block from $t = 0$ to $t = T$? (Assume $g$ is the acceleration due to gravity and let $C = \int_0^1 \cos\left(\frac{\pi}{2}x^2\right) dx$)

A uniform rod of cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear expansion $\alpha$ is initially at a uniform temperature of $10^\circ\text{C}$. It is placed horizontally on a smooth floor, wedged between two fixed rigid blocks. The faces of the blocks in contact with the rod are smooth inclined planes making an angle of $30^\circ$ with the horizontal. The ends of the rod are cut to perfectly match the $30^\circ$ incline. The rod is constrained by a smooth horizontal channel so that it cannot move vertically or bend. The ends of the rod are then maintained at steady temperatures of $90^\circ\text{C}$ and $50^\circ\text{C}$, while its lateral surface is thermally insulated. Assuming the blocks do not deform, what is the magnitude of the normal force $N$ exerted by each inclined plane on the rod in the steady state?

A wedge of mass $M = 3 \text{ kg}$ rests on a smooth horizontal floor. Its inclined surface makes an angle $\theta = 37^\circ$ with the horizontal and is oriented such that its outward normal vector points upwards and to the left. A block of mass $m = 1 \text{ kg}$ rests on the incline. A horizontal force $F(t) = 40t \text{ N}$ ($t$ in seconds) is applied to the wedge towards the left. The coefficient of static friction between the block and the wedge is $\mu_s = 0.5$. Find the work done by the static friction force on the block in the ground frame from $t = 0$ until the moment the block just begins to slip. (Take $g = 10 \text{ m/s}^2$, $\sin 37^\circ = 0.6$, $\cos 37^\circ = 0.8$)

A non-uniform cylinder of mass $m$ and radius $R$ has its center of mass at a distance $R/2$ from its geometric axis. Its moment of inertia about the geometric axis is $I_0 = m R^2$. The cylinder is placed on an inclined plane of angle $\theta = 45^\circ$. It is released from rest such that the line joining the geometric center to the center of mass is parallel to the incline and points downwards. What is the minimum coefficient of static friction $\mu_s$ required between the cylinder and the incline so that it begins to roll without slipping?

A uniform conducting rod of mass $m = 0.14 \text{ kg}$ and length $L = 0.5 \text{ m}$ rests symmetrically on two parallel, rough horizontal rails separated by a distance $L$. The rails are parallel to the y-axis and the rod is parallel to the x-axis. A uniform magnetic field $\vec{B} = B_0 (3\hat{i} + 4\hat{j} + 5\hat{k})$ exists in the region, where $B_0 = 0.1 \text{ T}$. The coefficient of static friction between the rod and the rails is $\mu = 0.5$. If a current $I$ is passed through the rod in the $+x$ direction, what is the minimum current $I$ required to just initiate sliding of the rod? (Take $g = 10 \text{ m/s}^2$)

A block of mass $m = 5 \text{ kg}$ and variable internal density (configured such that its center of mass is at its base to prevent toppling) rests on a fixed rough wedge of angle $\theta = 37^\circ$. The block is connected to a hanging mass $M = 4 \text{ kg}$ via a light inextensible string passing over a frictionless pulley at the top of the wedge, with the string parallel to the incline. The coefficient of static friction between the block and the wedge is $\mu = 0.5$. Initially, the system is in static equilibrium. Water of density $\rho_w = 1000 \text{ kg/m}^3$ slowly fills the container holding the wedge. Find the submerged volume of the block (in liters) at the exact moment it begins to slip up the incline. (Take $g = 10 \text{ m/s}^2$, $\sin 37^\circ = 0.6$, $\cos 37^\circ = 0.8$)

A uniform solid cylinder of mass $M$ and radius $R$ rolls without slipping on a rough horizontal plane with a coefficient of static friction $\mu$. A torsional spring of stiffness $\kappa$ is attached to its central axis, providing a restoring torque $\tau = -\kappa \theta$ proportional to the angle of rotation $\theta$, without exerting any net horizontal force. Determine the maximum angular amplitude $\theta_{\text{max}}$ of the oscillation such that the cylinder never slips.

A block of mass $m = 2 \text{ kg}$ and positive charge $q = 4 \text{ mC}$ is released from rest at $x=0$ on a rough inclined plane making an angle $\theta = 37^\circ$ with the horizontal. The coefficient of friction between the block and the plane is $\mu = 0.5$. A non-uniform electric field exists in the region, directed perpendicular to the inclined plane and pointing into the surface. The magnitude of this electric field varies with the distance $x$ down the incline as $E = kx$, where $k = 2.5 \times 10^3 \text{ V/m}^2$. Taking $g = 10 \text{ m/s}^2$, what is the maximum kinetic energy attained by the block during its downward motion?

A wedge has a rough inclined surface that makes an angle $\theta = 53^\circ$ with the horizontal, sloping downwards from left to right. A block of mass $m$ is placed on the incline. The coefficient of kinetic friction between the block and the incline is $\mu_k = 0.25$. The wedge is given a constant horizontal acceleration $a = 5 \text{ m/s}^2$ to the right. Assuming the block is released from rest relative to the wedge, what is the magnitude of the block's acceleration relative to the wedge? (Take $g = 10 \text{ m/s}^2$, $\sin 53^\circ = 0.8$, $\cos 53^\circ = 0.6$)

A block of mass $m = 2 \text{ kg}$ and density $\rho = 8000 \text{ kg/m}^3$ is released from rest on a rough inclined plane of angle $\theta = 37^\circ$. The entire setup is submerged in a large tank containing a viscous fluid of density $\sigma = 2000 \text{ kg/m}^3$. The coefficient of kinetic friction between the block and the plane is $\mu = 0.25$. The fluid exerts a drag force $F_d = k v^2$ on the block, where $k = 0.6 \text{ kg/m}$ and $v$ is the instantaneous velocity. Assuming the fluid is deep enough, what is the terminal velocity of the block? (Take $g = 10 \text{ m/s}^2$, $\sin 37^\circ = 3/5$)

A block of mass $m = 1 \text{ kg}$ slides down a rough inclined plane of angle $\theta = 37^\circ$. The coefficient of kinetic friction between the block and the incline is $\mu = 0.5$. At the bottom of the incline, the block strikes a smooth vertical plate attached to a massless spring of force constant $k = 25 \text{ N/m}$. The spring is oriented horizontally and is initially unstretched. If the speed of the block just before impact is $v_0 = 3\sqrt{2} \text{ m/s}$, find the maximum distance $s$ the block moves along the incline while compressing the spring. (Take $g = 10 \text{ m/s}^2$ and $\sin 37^\circ = 3/5$)

Two identical small spheres, each of mass $m$ and charge $q$, are suspended from a common point by two insulating, inextensible strings of length $L$. The system is placed inside a closed container completely filled with a dielectric liquid of dielectric constant $K$. The density of the material of the spheres is $\rho$ and the density of the liquid is $\rho/2$. The container is given a constant horizontal acceleration $a$. Assuming the angle between the strings remains very small, what is the steady-state equilibrium separation $x$ between the two spheres?

A uniform rigid rod of mass $2m$ and length $L$ rests on a smooth horizontal surface. A particle of mass $m$ moving with a horizontal velocity $v_0$ perpendicular to the rod collides perfectly inelastically with the rod at a distance of $\frac{L}{4}$ from its center. What is the angular velocity $\omega$ of the system immediately after the collision?

A uniform thin spherical shell of mass $M$ and radius $R$ is initially at rest in free space. A point mass $m$ is released from rest at an infinite distance and moves towards the shell along a straight line passing through its center. The shell has a tiny hole that allows the point mass to enter its interior. Let $v_{\text{rel}}$ be the relative speed of the point mass with respect to the shell when it is at a distance $R/2$ from the center. Let $\Phi$ be the magnitude of the net gravitational flux through a spherical surface of radius $2R/3$ centered at the shell's center, at this same instant. Which of the following represents the pair $(v_{\text{rel}}, \Phi)$?

A uniform thick rope of mass $M$, length $L$, cross-sectional area $A$, and Young's modulus $Y$ is placed on a smooth horizontal surface. A constant horizontal force $F$ is applied at one end to accelerate it along its length. Let $\Delta L$ be the total elongation of the rope in steady state (assuming $\Delta L \ll L$), and $t_0$ be the time taken by a small transverse wave pulse to travel from the free end to the pulled end. Which of the following expressions represents the ratio $\frac{\Delta L}{t_0^2}$?

A wedge of mass $M = 3 \text{ kg}$ and inclination angle $\theta = 45^\circ$ is placed on a smooth horizontal surface. A block of mass $m = 2 \text{ kg}$ is placed on the smooth inclined surface of the wedge. The system is released from rest. Taking the acceleration due to gravity as $g = 10 \text{ m/s}^2$, what is the magnitude of the normal force exerted by the wedge on the block during the motion?

An elevator accelerates downwards with a constant acceleration $a = \frac{g}{3}$. Inside the elevator, a tall cylindrical tank filled with a viscous liquid is fixed to the floor. A small sphere of mass $m$ is released from rest at the surface of the liquid. The liquid exerts a drag force $F_d = -kv$ on the sphere, where $v$ is the velocity of the sphere relative to the liquid. Neglecting buoyant force, what is the distance fallen by the sphere relative to the elevator at the instant the apparent weight of the sphere (defined here as the magnitude of the drag force exerted by the liquid) reaches half of its maximum possible value?

Two fixed concentric uniform spherical shells have masses $M$ and $2M$, and radii $R$ and $2R$ respectively. An elevator is located in the region between the shells at a radial distance $r$ ($R < r < 2R$) from their common center. The absolute gravitational potential at the elevator's location is $V(r)$ (taking $V \to 0$ at infinity). The elevator is programmed to move such that its radial outward acceleration is exactly $a = \frac{V(r)}{r}$. A test mass $m$ is suspended from a spring balance inside the elevator. What is the reading of the spring balance when the elevator is at $r = \frac{3R}{2}$?

An elevator is accelerating vertically upwards with an acceleration $a_e = 5 \text{ m/s}^2$. Inside the elevator, a smooth wedge of mass $M = 4 \text{ kg}$ and inclination $\theta = 30^\circ$ rests on the horizontal frictionless floor. A small block of mass $m = 2 \text{ kg}$ is placed on the inclined surface of the wedge and released from rest relative to the wedge. What is the magnitude of the normal force exerted by the wedge on the block? (Take $g = 10 \text{ m/s}^2$)

A small block of mass $m$ and charge $+q$ is constrained to move with a constant speed $v$ along a smooth circular groove of radius $R$ cut into the horizontal floor of an elevator. The elevator accelerates upwards with a constant acceleration $a$. A uniform magnetic field $\vec{B} = B_0 \hat{i} + \sqrt{3}B_0 \hat{j} + 2B_0 \hat{k}$ exists in the region, where $\hat{i}$ and $\hat{j}$ are unit vectors in the horizontal plane and $\hat{k}$ is vertically upwards. Assuming the block never loses contact with the floor, what is the time-averaged apparent weight of the block (the vertical normal force exerted by the floor) over one complete revolution?

A rigid block of mass $M$ rests on a weighing scale on the floor of an elevator. The block is also attached to the rigid ceiling of the elevator by a vertical metallic wire. At an initial temperature $T_0$, the elevator is at rest, the wire has its exact unstretched length $L_0$, cross-sectional area $A_0$, and Young's modulus $Y_0$. The tension in the wire is initially zero. The elevator then accelerates upwards with a constant acceleration $a$. Simultaneously, the temperature of the wire is decreased by $\Delta T$. The coefficient of linear expansion of the wire is $\alpha$, and its Young's modulus varies with temperature $T$ as $Y = Y_0[1 - \beta(T - T_0)]$. Assuming the distance between the ceiling and the scale remains strictly $L_0$, and $\alpha \Delta T \ll 1$, $\beta \Delta T \ll 1$, what is the apparent weight of the block (the normal force recorded by the scale)?

A block of mass $1 \text{ kg}$ rests on a rough horizontal surface. The coefficients of static and kinetic friction between the block and the surface are $\mu_s = 0.6$ and $\mu_k = 0.4$, respectively. A time-varying horizontal force $F(t) = 6t$ (in Newtons, with $t$ in seconds) is applied to the block starting at $t = 0$. What is the kinetic energy of the block after it has moved a distance of $2 \text{ m}$ from its initial position? (Take $g = 10 \text{ m/s}^2$)

A thin spherical shell of radius $R$ is fixed in deep space. Its surface mass density varies as $\sigma(\theta) = \sigma_0 (1 - \cos \theta)$, where $\theta \in [0, \pi]$ is the polar angle measured from the positive Z-axis. A small block of mass $m$ rests on a rough flat surface (coefficient of static friction $\mu$) fixed exactly at the center of the shell. The surface lies in the X-Y plane. What is the minimum force $F$ required to just move the block?

A small block of mass $m$ and positive charge $q$ is projected up a rough inclined plane with an initial velocity $v_0$. The angle of inclination of the plane is $\theta$ and the coefficient of kinetic friction between the block and the plane is $\mu$. A uniform magnetic field of magnitude $B$ exists in the region. The magnetic field is directed parallel to the incline surface and perpendicular to the line of greatest slope, such that the resulting magnetic force pushes the block perpendicularly into the incline. Acceleration due to gravity is $g$. Which of the following expressions correctly represents the maximum distance $d$ traveled by the block up the incline before coming to rest momentarily?

A solid cylinder of mass $m = 2 \text{ kg}$ and radius $R$ is placed on a rough inclined plane inclined at an angle $\theta = 60^\circ$ to the horizontal. The coefficient of static friction between the cylinder and the plane is $\mu = \frac{1}{2\sqrt{3}}$. A force $F$ is applied to the center of mass of the cylinder, directed upwards along the inclined plane. What is the minimum magnitude of $F$ required to ensure that the cylinder rolls down the incline without slipping? (Take $g = 10 \text{ m/s}^2$)

A uniform rigid rod of mass $m$ and length $L$ is initially at rest on a smooth horizontal surface. An impulsive force of magnitude $J$ is applied at one end of the rod. The direction of the impulse makes an angle $\theta = \frac{\pi}{6}$ with the longitudinal axis of the rod. What is the distance of the instantaneous center of rotation (ICR) of the rod from its center of mass immediately after the impulse?

A particle of mass $m$ and charge $q > 0$ is projected from the origin $(0,0,0)$ with an initial velocity $\vec{v} = v_0 \hat{i}$. A spatially varying magnetic field $\vec{B} = B_0 x \hat{k}$ exists in the region $x > 0$, where $B_0$ is a positive constant. Neglecting gravity, what is the maximum $x$-coordinate ($x_{\text{max}}$) reached by the particle?

A nucleus $A$ of mass $6m$ is initially at rest in the laboratory frame. It undergoes a radioactive decay into a nucleus $B$ of mass $4m$ and a particle $C$ of mass $2m$, releasing a total kinetic energy of $4Q$. Subsequently, while in flight, nucleus $B$ decays into a nucleus $D$ of mass $3m$ and a particle $E$ of mass $m$, releasing a total kinetic energy of $Q$. Assuming all decays are isotropic in their respective center-of-mass frames and non-relativistic mechanics applies, what is the maximum possible kinetic energy of nucleus $D$ in the laboratory frame?

A steady, incompressible fluid of density $\rho$ flows out of a circular pipe of radius $R$. The velocity profile of the fluid across the pipe's cross-section is non-uniform and given by $v(r) = v_0 \left(1 - \frac{r^2}{R^2}\right)$, where $r$ is the radial distance from the central axis and $v_0$ is the maximum velocity. The fluid stream strikes a stationary hemispherical deflector placed coaxially, which completely reverses the direction of the fluid flow ($180^\circ$ deflection). Assuming the fluid leaves the deflector with the same speed it had upon striking for each streamline, what is the magnitude of the horizontal force exerted by the fluid on the deflector?

A thermally insulated vessel of mass $M$ contains a monoatomic ideal gas of total mass $m_0$ (where $m_0 \ll M$) and molar mass $\mu$ at an initial temperature $T_0$. The vessel is initially at rest in gravity-free space. A small hole of area $A$ is opened, allowing the gas to effuse out. The thrust force experienced by the vessel due to effusion is $F = \frac{1}{2} P A$, where $P$ is the instantaneous pressure inside the vessel. Assuming each effusing molecule carries away an average kinetic energy of $2kT$, what is the final speed of the vessel after all the gas has escaped?

A rigid container of negligible mass ($M_{\text{container}} \approx 0$) is initially filled with a monatomic ideal gas of total mass $M_0$ and temperature $T_0$. The container is initially at rest in deep space. A small valve is opened, and the gas is expelled to the vacuum through a perfectly designed nozzle such that the exit pressure of the gas is zero. Assume the gas remaining in the container undergoes a reversible adiabatic process, and the flow through the nozzle is steady and isentropic. Find the velocity of the container when the mass of the gas remaining inside is $\frac{M_0}{8}$. (Let $R$ be the universal gas constant and $\mu$ be the molar mass of the gas).

A block of initial mass $m_0 = 10 \text{ kg}$ rests on a rough inclined plane angled at $\theta = 30^\circ$ to the horizontal. The coefficient of static friction is $\mu_s = 1/\sqrt{3}$. The block is equipped with a pulse-jet that expels gas at a constant rate $\mu = 0.1 \text{ kg/s}$ with a relative exhaust velocity $u = 20\sqrt{3} \text{ m/s}$. The jet provides a thrust force whose line of action is in the vertical plane containing the line of greatest slope, directed upwards and outwards at an angle $\alpha = 30^\circ$ relative to the incline. At what time $t$ does the block just begin to slide up the incline? (Take $g = 10 \text{ m/s}^2$)