System of Particles and Rotational Motion - Class 11 Part 1

A gyroscope consists of a uniform solid disk of mass $M = 2 \text{ kg}$ and radius $R = 0.1 \text{ m}$ mounted on a massless axle. The axle is attached to a frictionless pivot on the vertical axis of a rotating platform. The distance from the pivot to the center of mass of the disk is $L = 0.2 \text{ m}$. The platform rotates at a constant angular velocity $\Omega = 10 \text{ rad/s}$ about the vertical axis. The gyroscope axle is observed to maintain a steady angle $\theta = 60^\circ$ with the upward vertical, precessing together with the platform. Calculate the required spin angular velocity $\omega$ of the disk about its axle. (Take $g = 10 \text{ m/s}^2$)

A physical pendulum consists of a uniform thin rod of mass $m$ and length $R$, and a uniform solid circular disc of mass $m$ and radius $R$. The disc is rigidly attached at its center to one end of the rod, such that they lie in the same plane. The system is suspended from a smooth horizontal pivot passing through the other end of the rod, perpendicular to their plane, and performs small oscillations. The coefficient of linear expansion of the rod is $\alpha_1$ and that of the disc is $\alpha_2$. If the temperature of the system is increased by a small amount $\Delta\theta$, the fractional change in the time period $\frac{\Delta T}{T}$ is:

A solid sphere of mass $M$ and radius $r$ rolls without slipping on the inside surface of a fixed hollow cylinder of radius $R$ ($R > r$). The motion is planar, and $\theta$ represents the small angular displacement of the sphere's center from the vertical. A constant rolling resistance torque $\tau_0$ opposes the rotation of the sphere about its own center. If the sphere is released from rest at an initial angle $\theta_0$ (where $\theta_0 > \frac{\tau_0}{Mgr}$), what is the square of the maximum angular velocity ($\dot{\theta}_{max}^2$) of the sphere's center during its first downward swing?

A uniform solid disc of mass $m = 2 \text{ kg}$ and radius $R = 0.5 \text{ m}$ is pivoted at a point on its edge. The pivot is attached to a straight torsional wire with a torsional constant $C = 7 \text{ N m rad}^{-1}$. The wire is perpendicular to the plane of the disc and serves as the axis of rotation. The setup is fixed such that the wire makes an angle of $30^\circ$ with the downward vertical. If the disc is given a small angular displacement in its own plane from its stable equilibrium position, what is the time period of the resulting simple harmonic motion? (Assume $g = 10 \text{ m s}^{-2}$)

A heavy parent nucleus is modeled as a rigid thin rod of length $L$ with a non-uniform linear mass density $\lambda(x) = \lambda_0\left(1 + \frac{2x}{L}\right)$ for $0 \le x \le L$. It is initially rotating with a constant angular velocity $\omega$ about an axis perpendicular to the rod passing through its geometric center, which is at rest in the laboratory frame. The nucleus undergoes fission by splitting exactly at its geometric center into two fragments. The binding energy per unit mass of the parent nucleus is $B_1$ and that of both fragments is $B_2$ ($B_2 > B_1$). Assume all available energy in the center-of-mass (COM) frame is converted entirely into the relative translational kinetic energy of the fragments. What is the relative speed $v_{rel}$ of the two fragments immediately after fission?

A system consists of a heavy base of mass $M$ resting on the horizontal floor of an elevator. A motor rigidly attached to the base drives a massless rod of length $L = 0.5 \text{ m}$ with a point mass $m = 2 \text{ kg}$ at its free end. The rod rotates in a vertical plane with a constant angular velocity $\omega = 10 \text{ rad/s}$. The elevator accelerates upwards with a constant acceleration $a = 5 \text{ m/s}^2$. Assume $g = 10 \text{ m/s}^2$. What is the maximum internal torque $\tau_{\text{max}}$ the motor must provide to maintain the constant angular velocity, and what is the minimum mass $M_{\text{min}}$ of the base required so that it never loses contact with the elevator floor?

A rigid rod of length $L$ has a non-uniform linear mass density given by $\lambda = kx^2$, where $x$ is the distance from its left end and $k$ is a positive constant. The rod is pivoted at its geometric center ($x = L/2$) and is free to oscillate in a vertical plane under the acceleration due to gravity $g$. What is the time period $T$ of small oscillations?

A microscopic cavity contains $N$ molecules of an ideal rigid diatomic gas, each of mass $m$. The cavity is driven by an external mechanism to oscillate along the $x$-axis with a velocity $v(t) = V_0 \cos(\omega t)$. The frequency $\omega$ is sufficiently high such that the center of mass of the gas oscillates synchronously with the cavity. In a dynamic steady state, the time-averaged total kinetic energy of the gas (including both translational and rotational motion) measured in the stationary laboratory frame is $E$. Which of the following expressions gives the true thermodynamic temperature $T$ of the gas? (Here $k_B$ is the Boltzmann constant).

A rigid cylindrical tube of mass $M$ and length $L$ is initially at rest in deep space, free from any external forces. Inside the tube, a non-uniform liquid of mass $M$ initially occupies the left half of the tube ($0 \le x \le L/2$, where $x$ is the distance from the left end of the tube). The linear mass density of the liquid is given by $\lambda(x) = \lambda_0 \sin\left(\frac{\pi x}{L}\right)$. An internal pump transfers the entire liquid to the right half of the tube ($L/2 \le x \le L$), where it settles with the density profile $\lambda(x) = \lambda_0 \sin\left(\frac{\pi x}{L}\right)$. What is the magnitude of the absolute displacement of the tube?

A very long uniform plank of mass $M$ lies on a smooth horizontal surface. A small block of mass $m$ is initially at rest at the geometric center of the plank. At $t=0$, an internal mechanism causes the block to move along the longitudinal axis of the plank such that its velocity relative to the plank is $v_{rel}(t) = c t$, where $c$ is a positive constant. Simultaneously, an external time-varying force of magnitude $F(t) = kt$ (where $k$ is a positive constant) is applied to the geometric center of the plank. The direction of this force is continuously adjusted so that it always remains perpendicular to the longitudinal axis of the plank. Find the magnitude of the angular velocity of the plank as $t \to \infty$.

Two blocks, A and B, of masses $2m$ and $m$ respectively, are connected by a massless spring of spring constant $k$. They are placed on a horizontal surface such that block A is on a rough patch with a coefficient of kinetic friction $\mu$, while block B is on a smooth patch. Initially, the spring is at its natural length and the entire system is given an initial velocity $v_0$ in the direction from A to B. Assuming $v_0$ is sufficiently large so that block A does not stop before the spring reaches its maximum extension, what is the maximum extension of the spring?

A uniform rod of mass $M$ (in kg) and length $2L$ (in m) is suspended horizontally at its center by a vertical torsion wire of torsional constant $C$ (in N$\cdot$m/rad). Two small collars, each of mass $m$ (in kg), slide symmetrically outward along the rod. At a particular instant, the rod has an angular displacement $\theta$ (in rad) from its equilibrium position and an angular velocity $\omega$ (in rad/s) in the same sense. At this instant, the collars are at a distance $x$ (in m) from the center and are moving outward with a speed $v$ (in m/s) relative to the rod. Which of the following expressions gives the algebraic tangential force $F_t$ (in N) exerted by the rod on one of the collars in the direction of the angular velocity?

A radioactive nucleus, initially at rest in a gravity-free space, undergoes continuous decay by emitting a stream of particles in a fixed direction. The speed of the emitted particles relative to the nucleus is a constant $u$. The mass of the nucleus decreases according to the radioactive decay law $m(t) = m_0 e^{-\lambda t}$, where $m_0$ is the initial mass and $\lambda$ is the decay constant. Which of the following expressions gives the distance $x(t)$ traveled by the nucleus as a function of time $t$?

A spacecraft of initial mass $m_0$ is equipped with a perfectly absorbing solar sail of area $A$. It is propelled by a constant radiation pressure $P$ from a distant star, which exerts a force $F = PA$ on the sail. The spacecraft travels along a straight line through a stationary dust cloud of uniform mass density $\rho$. Assuming all dust striking the sail sticks to it and the spacecraft starts from rest at $t=0$, which of the following expressions gives the velocity $v$ of the spacecraft as a function of time $t$?

A small droplet of initial mass $m_0$ falls from rest under gravity through a stationary cloud. As it falls, it accumulates mass from the cloud such that its mass at a vertical distance $x$ from the release point is given by $m(x) = m_0\left(1 + \frac{x}{H}\right)$, where $H$ is a positive constant. Assuming no resistive forces other than the momentum transfer from the accumulated stationary mass, what is the speed of the droplet after it has fallen a distance $H$?

A long capillary tube of radius $r$ is held vertically such that its lower end just touches the surface of a large pool of liquid of density $\rho$ and surface tension $T$. The contact angle of the liquid with the tube is zero. The entire setup is placed in an elevator which accelerates upwards with a constant acceleration $a$. Assuming the liquid flow is non-viscous and neglecting the kinetic energy of the liquid outside the tube, what is the maximum height $H$ reached by the liquid column in the capillary tube relative to the liquid level in the pool?

A uniform horizontal circular disc of mass $M$ (in kg) and radius $R$ (in m) is suspended by a vertical wire of torsional constant $K$ (in N$\cdot$m/rad). Initially, it is undergoing torsional oscillations. At time $t=0$ (in s), the disc passes through its mean position (angular displacement $\theta = 0$ rad) with angular velocity $\omega_0$ (in rad/s). From $t=0$ onwards, sand falls vertically onto the edge of the disc at a constant rate $\mu = \frac{dm}{dt}$ (in kg/s) and sticks to it. Assuming the sand has no initial horizontal velocity, which of the following expressions gives the angular acceleration $\alpha$ (in rad/s$^2$) of the disc at any time $t > 0$ in terms of its instantaneous angular displacement $\theta$ and angular velocity $\omega$?

A uniform rod of mass $m$ and length $l$ is suspended vertically from a hinge $A$. The hinge is attached to a block of mass $M = 2m$ which can slide without friction along a horizontal overhead rail. Initially, the system is at rest. A horizontal impulse $J$ is applied to the bottom end $B$ of the rod. Find the angular velocity $\omega$ of the rod immediately after the impulse.

A uniform solid cylinder of mass $M = 2m$ and radius $R$ rests on its flat face on a smooth horizontal table. A uniform thick rope of mass $m$ and length $L$ is completely wrapped around the cylinder. A constant horizontal force $F$ is applied to the free end of the rope, causing it to unwrap. The unwrapped portion of the rope remains straight and tangent to the cylinder. At the instant when exactly half the length of the rope ($L/2$) is unwrapped, the cylinder has an angular velocity $\omega$. What is the angular acceleration $\alpha$ of the cylinder at this instant?

A uniform rigid rod of mass $m$ and length $L$ is hinged freely at one end $O$. A light elastic spring of spring constant $k$ and natural length $L$ has one end attached to the free end $A$ of the rod. The other end of the spring is fixed to a point $P$ located at a distance $2L$ directly below the hinge $O$. The rod is held horizontal and released from rest. Find the angular velocity $\omega$ of the rod when it passes through the lowest vertical position. (Here $g$ is the acceleration due to gravity).

A block of mass $M$ rests on a smooth horizontal surface. A uniform rod of mass $m$ and length $L$ is hinged at its upper end $A$ to the block. The lower end $B$ of the rod is connected to a fixed vertical wall to the right by a horizontal massless inextensible string. Initially, the rod is vertical and the string is taut. A constant horizontal force $F$ is applied to the block towards the right. Find the tension in the string immediately after the force is applied.

A rigid circular loop of mass $m$ and radius $R$ is made of a uniform thin wire carrying a steady current $I$. The loop is placed in a gravity-free space with a uniform magnetic field $\vec{B}$ of magnitude $B_0$ present throughout. Initially, the loop is held at rest such that its magnetic dipole moment vector is perpendicular to the direction of $\vec{B}$. The loop is then released and is free to rotate about its center of mass. Assume the current $I$ is maintained constant by an external source, and ignore self-inductance. What is the maximum angular speed acquired by the loop?

A small block of mass $m = 1 \text{ kg}$ is placed on a smooth horizontal table and connected to one end of a spring with natural length $l_0 = 1 \text{ m}$ and spring constant $k = 2 \text{ N/m}$. The other end of the spring is fixed to a point $O$ on the table. The spring exhibits an elastic after-effect, exerting an internal non-conservative damping force $F_d = -b v_r$ where $v_r$ is the radial velocity of the block. Initially, the block is at a distance $r_0 = 4 \text{ m}$ from $O$ and is given a purely transverse velocity $v_0 = 1 \text{ m/s}$. Due to the damping force, the radial motion eventually ceases, and the block settles into a steady circular orbit around $O$. Find the total work done by the non-conservative damping force during this entire process.

A uniform solid cylinder of mass $m$ and radius $R$ rests on a rough horizontal surface with a coefficient of friction $\mu = \frac{1}{4\sqrt{3}}$. A sharp impulse of magnitude $J$ is applied to the cylinder at its topmost point. The line of action of the impulse is directed downwards at an angle $\theta = 60^\circ$ below the horizontal. What is the angular velocity of the cylinder immediately after the impulse?

A thin rod is composed of two uniform cylindrical halves of equal mass $m$ and initial length $L_0$ each. The coefficients of linear expansion of the left and right halves are $\alpha_1$ and $\alpha_2$ respectively ($\alpha_1 > \alpha_2$). The rod is placed inside a fixed smooth spherical cavity of radius $R$ ($R > L_0$). Initially, at temperature $T_0$, the rod rests horizontally in stable equilibrium. If the temperature of the system is increased by a small amount $\Delta T$, the rod tilts to a new equilibrium position. Assuming $\Delta T$ is small enough that higher-order terms can be neglected, what is the angle $\theta$ the rod makes with the horizontal?

A uniform rigid rod of mass $M$ and length $L$ is placed in a smooth straight radial groove on a horizontal turntable. The turntable rotates at a constant angular velocity $\omega$ about a vertical axis passing through its center. Initially, the rod is held at rest relative to the turntable with its inner end at a distance $r_0$ from the rotation axis. The rod is then released. What is the total work done by the normal force exerted by the groove on the rod in the inertial frame of reference, from the moment of release until its inner end reaches a distance $2r_0$?

A uniform rod of mass $M$ and length $L$ is pivoted at a frictionless hinge $O$ located at a distance $L/4$ from its center of mass. The rod is released from rest at an initial angular displacement $\theta_0$ from the vertical downward position. Since $\theta_0$ is large, the small-angle approximation ($\sin\theta \approx \theta$) is invalid. Let $g$ be the acceleration due to gravity. Which of the following expressions represents the exact time period $T$ of the resulting oscillations?

A solid sphere of mass $m$ and radius $R$ rolls without slipping down a rough inclined plane that makes an angle $\theta_1 = 60^\circ$ with the horizontal. It encounters a sharp concave corner and immediately transitions onto a second inclined plane making an angle $\theta_2 = 30^\circ$ with the horizontal. The second inclined plane is completely frictionless. Assuming the collision at the corner is perfectly inelastic in the normal direction (the sphere does not bounce off the second plane), what fraction of the sphere's total kinetic energy is lost during the collision at the corner?

A uniform solid disk of original mass $9m$ and radius $R$ has a circular hole of radius $R/3$ cut out. The hole is centered at a distance $2R/3$ from the geometric center $O$ of the disk. The remaining non-uniform disk (mass $8m$) is suspended by a light inextensible string wrapped around its outer rim. The string passes over an ideal fixed pulley at the edge of a smooth horizontal table and is attached to a block of mass $4m$ resting on the table. At $t=0$, the system is released from rest with the hole's center located exactly to the right of $O$ at the same horizontal level. The string unwinds vertically from the left edge of the disk. Find the initial acceleration of the block.

A thin insulating disk of radius $R$ lies in the $xy$-plane with its center at the origin. It has a non-uniform mass surface density $\sigma_m(r) = \sigma_0 \left(1 - \frac{r}{R}\right)$ and a non-uniform charge surface density $\sigma_q(r) = \sigma_1 \frac{r}{R}$, where $r$ is the radial distance from the center. The disk is rotated with a constant angular velocity $\omega$ about a chord that lies in the $xy$-plane at a perpendicular distance of $R/2$ from the center. Find the ratio of the magnitude of the magnetic dipole moment to the angular momentum of the disk, both measured about this axis of rotation.

A rigid square loop of uniform thin wire has mass $m$, side length $a$, and self-inductance $L$. Its electrical resistance is negligible. The loop is initially at rest in the $xy$-plane in a gravity-free space. A uniform magnetic field $\mathbf{B} = B\hat{k}$ is present in the region. The loop is constrained to rotate about a fixed axis lying in the $xy$-plane. This axis is parallel to one of the loop's diagonals and passes through one of its vertices. An impulsive torque is applied, giving the loop an initial angular velocity $\omega_0$. Assuming the initial current in the loop is zero, what is the minimum value of $\omega_0$ required for the loop to complete a full rotation?

A solid cylinder of mass $m$ and radius $r$ rolls without slipping inside a fixed vertical circular track of radius $R = 5r$. The track exerts a constant rolling friction torque $\tau_r = \frac{1}{\pi} m g r$ on the cylinder, opposing its rotation. To just complete the vertical loop without losing contact at the highest point (vertex), the cylinder must be given a minimum initial velocity $v_0$ at the lowest point. Assuming rolling without slipping is maintained throughout, find $v_0$.

A U-tube of uniform cross-section has a horizontal base of length $2L$ and two vertical arms. The horizontal base is completely filled with a non-homogeneous liquid whose density varies linearly from $\rho_0$ at the left end to $3\rho_0$ at the right end. The vertical arms contain a homogeneous liquid of density $2\rho_0$. Massless, freely moving pistons separate the two liquids at the junctions of the base and the arms. The entire U-tube is rotated with a constant angular velocity $\omega$ about a vertical axis located at a distance $d$ to the left of the left end of the horizontal base. In the steady state, what is the difference in height $\Delta h$ between the liquid levels in the right and left vertical arms?

A solid uniform sphere of mass $M$ and radius $R$ is released from rest at a vertical height $h$ on a rough inclined plane. It rolls without slipping down the incline. At the exact instant it reaches the horizontal surface at the bottom, its center of mass is suddenly clamped by a fixed, frictionless pivot, instantly halting its translational motion but allowing it to rotate freely.

Simultaneously, a rigid, massless rod of length $R$, which is attached to the pivot and has a point mass $m = \frac{2}{5}M$ at its free end, is coupled to the sphere via a clutch. Initially, the rod was at rest, hanging vertically downwards. The clutch instantly forces the rod and the sphere to rotate together with the same angular velocity.

Assuming no external torques act during the coupling process, what is the maximum vertical height $d$ reached by the point mass $m$ as the system swings upwards?

A truck accelerates horizontally to the right with a constant acceleration $a = \frac{g}{2}$. Inside the truck, a fixed inclined plane makes an angle $\theta = 45^\circ$ with the horizontal floor. The incline slopes downward to the right (i.e., in the direction of the truck's acceleration). A solid sphere of mass $m$ and radius $R$ is released from rest on the incline. What is the minimum coefficient of static friction $\mu$ between the sphere and the incline required for the sphere to undergo pure rolling?

A solid cylinder of mass $m$ and radius $R$ is rolling without slipping down a rough inclined plane of angle $\theta = 30^\circ$ with the horizontal. It encounters a small fixed step of height $h = R/2$ perpendicular to the incline. Assuming the cylinder pivots about the edge of the step without slipping or bouncing during and after the impact, what is the minimum velocity $v_0$ of the center of mass just before impact so that the cylinder can just climb over the step? (Let $g$ be the acceleration due to gravity)

A solid sphere of radius $R$ has a non-uniform mass density given by $\rho(r) = \rho_0 \left(1 - \frac{r^2}{R^2}\right)$, where $r$ is the radial distance from its centre. The sphere is released from rest at a height $H = 1.8 \text{ m}$ on a rough inclined plane. It rolls down without slipping and smoothly transitions to a horizontal ledge at height $h = 5.0 \text{ m}$ above the ground. It then leaves the ledge horizontally and falls to the ground. Assuming acceleration due to gravity $g = 10 \text{ m/s}^2$ and no energy loss at the transition, what is the horizontal distance from the edge of the ledge to the point where the sphere hits the ground?

A rigid rectangular coil of area $A$ and constant resistance $R$ is constrained to rotate about the $z$-axis. A uniform, time-varying magnetic field $\vec{B}(t) = B_0 \sin(\omega_0 t) \hat{i}$ is present in the region. At $t=0$, the area vector of the coil is aligned with the positive $x$-axis ($\hat{n} = \hat{i}$). An external agent applies a torque to keep the coil rotating at a constant angular velocity $\vec{\omega} = \omega_0 \hat{k}$. What is the work done by the external agent during one complete revolution of the coil (from $t=0$ to $t=2\pi/\omega_0$)? Ignore self-inductance.

A diatomic molecule is modeled as a rigid massless rod of length $L$ with two point masses: $m_1 = m$ with charge $+q$, and $m_2 = 2m$ with charge $-q$. The molecule lies on a smooth horizontal insulating table inside a box. Mass $m_1$ is hinged to the floor of the box. A uniform electric field $\vec{E} = E \hat{i}$ exists throughout the space. The box is given a constant horizontal acceleration $\vec{a} = a \hat{i}$. Initially, the rod is held at rest (relative to the box) parallel to the positive $y$-axis. It is then released. Find the magnitude of the force exerted by the hinge on $m_1$ in the $x$-direction at the instant the rod aligns with the negative $x$-axis.

A uniform solid cube of mass $m$ and side length $a$ is placed on a rough inclined plane of angle $\theta$. The coefficient of kinetic friction between the cube and the plane is $\mu$. By means of external forces, the cube is manipulated such that its leading lower edge slides down the incline at a constant velocity $v_0$, while the cube simultaneously topples forward about this edge at a constant angular velocity $\omega_0$. Assume $\omega_0$ is small enough that the cube never loses contact with the incline. What is the magnitude of the work done by the friction force on the cube during the time it rotates from its initial flat position to an angle of $\pi/4$?

A spherical dust cloud of mass $M$ and radius $R$ initially has a uniform density and rotates with an angular velocity $\omega_0$. It undergoes gravitational collapse, radiating away energy, until it settles into a new spherical shape of radius $R/2$. In this final state, the cloud has a non-uniform density profile given by $\rho(r) = \frac{c}{r}$ (where $c$ is a constant and $r$ is the distance from the center). Assuming no external torques act on the system, what is the net energy radiated away during this process?

A uniform conducting rod of mass $m$ and length $l$ is pivoted at one of its ends $O$ and can rotate freely in a horizontal plane. A uniform vertical magnetic field $B$ exists in the region. The free end of the rod slides on a fixed, smooth, perfectly conducting circular ring of radius $l$ centered at $O$. A resistor of resistance $R$ is connected between the pivot $O$ and the ring. The rod has negligible resistance. At $t = 0$, an angular impulse $J$ is applied to the rod about the pivot. What is the total angle rotated by the rod before it comes to a complete stop?

A uniform rigid rod of mass $M$ and length $L$ is suspended vertically from a frictionless pivot at its upper end. A horizontal impulse $J$ is applied to the rod at its lower end, causing it to swing and reach a maximum angular displacement of $60^\circ$ from the vertical. If $g$ is the acceleration due to gravity, what is the magnitude of the impulsive reaction force exerted by the pivot on the rod during the strike?

A uniform solid cylinder of mass $M=4m$ and radius $R$ is mounted on a frictionless, fixed vertical axis. A projectile of mass $m$ is launched from the ground with an initial speed $u$ at an angle of $60^{\circ}$ to the horizontal. At the highest point of its trajectory, the projectile strikes the rough vertical surface of the cylinder. The vertical plane containing the projectile's trajectory is at a perpendicular distance $b = R/2$ from the cylinder's axis. The coefficient of restitution for the normal component of the collision is $e = 1/2$. If the projectile stops slipping on the cylinder during the infinitesimal duration of the collision, what is the minimum coefficient of kinetic friction $\mu$ between the projectile and the cylinder?

A uniform solid disk of mass $m$ and radius $R$ is initially rolling without slipping on a rough horizontal surface with a constant velocity $v_0$. An impulsive force of magnitude $J$ is applied at the center of mass of the disk. The impulse is directed downwards at an angle of $60^\circ$ below the horizontal, and its horizontal component is opposite to the direction of $v_0$. The coefficient of kinetic friction between the disk and the surface is $\mu = \frac{1}{6\sqrt{3}}$. What is the angular velocity of the disk immediately after the impulse? (Assume clockwise rotation is positive)

A rectangular block of width $b$ and height $h$ has a 2D density profile $\rho(x,y) = \rho_0 \left(\frac{x}{b} + \frac{y}{h}\right)$, where $x \in [0, b]$ and $y \in [0, h]$ are measured from the left and bottom faces respectively. The block is placed on a rough inclined plane with its bottom face resting on the incline and its left face pointing directly up the incline. As the angle of inclination $\theta$ is slowly increased, the block is observed to be on the verge of sliding and toppling simultaneously. What is the coefficient of static friction $\mu$ between the block and the incline?

A uniform cubical block of mass $m$ and side $a$ rests on a rough horizontal surface. The coefficients of static and kinetic friction between the block and the surface are $\mu_s = 0.4$ and $\mu_k = 0.3$, respectively. A horizontal force $F(t) = kt$ (where $k$ is a positive constant and $t$ is time) is applied horizontally at the top edge of the block. Find the magnitude of the linear momentum of the block at the exact instant it begins to topple. Assume acceleration due to gravity is $g$.

A uniform rectangular block of mass $m$, height $h$, and base width $b$ is released from rest on an inclined plane making an angle $\theta$ with the horizontal. A thin layer of fluid on the incline exerts a viscous drag force on the block of magnitude $F_v = kv$ directed up the incline, where $v$ is the instantaneous velocity of the block and $k$ is a positive constant. The drag force acts uniformly over the bottom face of the block. Given that $\tan\theta > \frac{b}{h}$, the block will eventually topple forward. Find the time $t$ after release at which the block begins to topple.

A uniform solid cylinder of radius $r$ and height $h=4r$ rests upright on a rough horizontal turntable. The center of its base is at a distance $R = 4 \text{ m}$ from the rotation axis. The turntable starts from rest at $t=0$. The arc length $s(t)$ traveled by the cylinder's base is recorded, and its $s-t$ graph is a parabola with a constant concavity $\frac{d^2s}{dt^2} = 3 \text{ m/s}^2$. The coefficient of static friction is $\mu = 0.8$. Taking $g = 10 \text{ m/s}^2$, determine the mode of instability (sliding or toppling) and the exact time $t$ it occurs.

A gyroscope consists of a thin uniform ring of initial mass $M_0$ and radius $R$, mounted on a massless shaft of length $l$. The shaft is pivoted at a fixed origin $O$. The ring spins about the shaft with an initial angular velocity $\omega_0$. A constant downward force $F$ is applied at the center of the ring (ignore gravity). The ring loses mass at a constant rate $\mu$ such that its mass at time $t$ is $M(t) = M_0 - \mu t$. The detached mass is ejected radially outwards relative to the ring's center. Assuming the rapid spin approximation ($\omega_0 \gg \Omega_p$), find the total angle of precession $\Delta \phi$ of the shaft from $t = 0$ to $t = \frac{M_0}{2\mu}$.