Angular Momentum & its Conservation

When the position vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ changes sign as $-\vec{r}$, which one of the following vector will not flip under sign change?

A particle of mass m is projected with velocity V making an angle of $45^\circ$ with the horizontal. The magnitude of the angular momentum of the projectile about an axis of projection when the particle is at maximum height h is ($g$ = acceleration due to gravity) $\left(\sin 45^\circ = \frac{1}{\sqrt{2}}\right)\ \left(\cos 45^\circ = \frac{1}{\sqrt{2}}\right)$

A particle of mass $m$ is projected with speed $v$ at $45^\circ$ with the horizontal. The magnitude of angular momentum of the particle about the point of projection when the particle is at its maximum height $h$ is $\left(\cos 45^\circ = \frac{1}{\sqrt{2}}\right)$

The position of an object having mass 0.1 kg as a function of time $t$ is given as

$\vec{r} = \left(10t^{2}\hat{i} + 5t^{3}\hat{j}\right)$ m. At $t = 1$ s, which of the following statements are correct? A. The linear momentum $\vec{p} = \left(2\hat{i} + 1.5\hat{j}\right)$ kg·m/s.

B. The force acting on the object $\vec{F} = \left(2\hat{i} + 3\hat{j}\right)$ N.

C. The angular momentum of the object about its origin $\vec{L} = 15\hat{k}$ J·s.

D. The torque acting on the object about its origin $\vec{\tau} = 20\hat{k}$ N·m.

Choose the correct answer from the options given below:

Two cars A and B each of mass $10^{3}$ kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is ________ J·s.

The moment of inertia of a ring about an axis passing through its centre and perpendicular to its plane is I. It is rotating with angular velocity $\omega$. Another identical ring is gently placed on it so that their centres coincide. If both rings are rotating about the same axis then loss in kinetic energy is

A thin horizontal disc is rotating about a vertical axis passing through its fixed centre $O$. Its angular momentum is $L_A$ and $L_B$ computed about points $A$ and $B$, respectively, with $OB = 2 \times OA$. The value of $\frac{L_A}{L_B}$ is:

Two small balls with masses $m$ and 2 m are attached to both ends of a rigid rod of length $d$ and negligible mass. If angular momentum of this system is $L$ about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about $A$ is:

A solid sphere isthrotating about a diameter at an angular velocity $\omega$. If it cools so that its radius reduces to $\left(\frac{1}{n}\right)$ of its original value, its angular velocity becomes

A kathak dancer is standing on horizontal surface with folded hands. In the begining dancer is rotating about his central axis and his kinetic energy is 'K' at that time. The kathak dancer now stretches his arms so that the moment of inertia of the dancer becomes three times and the angular velocity becomes one-third. The kinetic energy of the dancer now is

A mass tied to a string is whirled in a horizontal circular path with a constant angular velocity and its angular momentum is 'L'. If the length of the string is now halved, keeping angular velocity same then the angular momentum will be

A ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{M}{8}$ at rest at $O$. These masses can move radially outwards along two mass-less rods fixed on the ring as shown in figure. At some instant the angular speed of the system is $\frac{8\omega}{9}$ and one of the masses is at a distance of $\frac{3}{5}R$ from point $O$. At this instant, the distance of the other mass from point $O$ is

A disc of moment of inertia $I_1$ is rotating in horizontal plane about an axis passing through centre and perpendicular to its plane with constant angular speed $\omega_1$. Another disc of moment of inertia $I_2$ having zero angular speed is placed co-axially on a rotating disc. Now, both the discs are rotating with constant angular speed $\omega_2$. The energy lost by the entire system is $\frac{x\omega_1^2}{2}$ where the value of $x$ is

A uniform bar of length 12 cm and mass 20 m lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is:

The Sun rotates around its centre once in 27 days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence? Assume the Sun to be a sphere of uniform density.

If $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass ' $m$ ' having position vector as $\vec{r} = a\left(\hat{i}\cos\omega t + \hat{j}\sin\omega t\right)$. The direction of force is

The moment of inertia of a ring about an axis passing through its centre and perpendicular to its plane is I. It is rotating with angular velocity $\omega$. Another identical ring is gently placed on it so that their centres coincide. If both the rings are rotating about the same axis then loss in kinetic energy is

A man standing on a turn-table is rotating at a certain angular frequency with his arms outstretched. He suddenly folds his arms. If his moment of inertia with folded arms is $75\%$ of that with outstretched arms, then his rotational kinetic energy will

A solid sphere with uniform density and radius $R$ is rotating initially with constant angular velocity $(\omega_1)$ about its diameter. After some time during the rotation it starts losing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become $R/2$ is $x\omega_1$. The value of $x$ is _________.

Two discs of moment of inertia ' $I_1$ ' and ' $I_2$ ' and angular speeds ' $\omega_1$ ' and ' $\omega_2$ ' are rotating along the collinear axes passing through their centre of mass and perpendicular to their plane. If the two discs are made to rotate together along the same axis. The rotational kinetic energy of the system will be

The position vectors of two 1 kg particles, (A) and (B), are given by

$\vec{r}_A = \left(\alpha_1 t^2\hat{i} + \alpha_2 t\hat{j} + \alpha_3 t\hat{k}\right)\text{ m}$ and$\vec{r}_B = \left(\beta_1 t\hat{i} + \beta_2 t^2\hat{j} + \beta_3 t\hat{k}\right)\text{ m}$, respectively;

$\left(\alpha_1 = 1\text{ m/s}^2, \alpha_2 = 3n\text{ m/s}, \alpha_3 = 2\text{ m/s}, \beta_1 = 2\text{ m/s}, \beta_2 = -1\text{ m/s}^2, \beta_3 = 4p\text{ m/s}\right)$, where t is time, n and $p$ are constants. At and velocities $\vec{V}_A$ and $\vec{V}_B$ of the particles are orthogonal to each other. At $t = 1\text{ s}$, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{L}\text{ kg m}^{2}\text{ s}^{-1}$. The value of L is _________.

The earth is assumed to be a sphere of radius ' $R$ ' and mass ' $M$ ' having period of rotation ' $T$ '. The angular momentum of earth about its axis of rotation is

A particle of mass ' m ' is rotating in a circular path of radius ' $r$ '. Its angular momentum is ' $L$ ' The centripetal force acting on it is ' $F$ '. The relation between ' $F$ ', ' $L$ ', ' $r$ ' and ' $m$ ' is

A particle of mass $\mathrm{m}$ is projected with a velocity '$\mathrm{u}$' making an angle of $30^\circ$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $\mathrm{h}$ is:

A body of mass '$m$' is projected with a speed '$u$' making an angle of $45^\circ$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\frac{\sqrt{2}mu^{3}}{Xg}$. The value of '$X$' is _________.

A body of mass $5\text{ kg}$ moving with a uniform speed $3\sqrt{2}\text{ ms}^{-1}$ in $X - Y$ plane along the line $y = x + 4$. The angular momentum of the particle about the origin will be _________ $\text{kg m}^{2}\ \text{s}^{-1}$.

A circular table is rotating with an angular velocity of $\omega\text{ rad/s}$ about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of $1\text{ m}$ on the groove. All the surfaces are smooth. If the radius of the table is $3\text{ m}$, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is $x\sqrt{2}\,\omega\text{ m/s}$, where the value of $x$ is _________.

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be ________ hours 30 minutes.

A thin uniform circular disc of mass ' $M$ ' and radius ' $R$ ' is rotating with angular velocity ' $\omega$ ' in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same radius but of mass $\left(\frac{M}{3}\right)$ is placed gently on the first disc co-axially. The new angular velocity will be

A thin circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. If another disc of same dimensions but of mass $M/2$ is placed gently on the first disc co-axially, then the new angular velocity of the system is:

Two discs of moment of inertia $I_1 = 4\text{ kg m}^{2}$ and $I_2 = 2\text{ kg m}^{2}$, about their central axes & normal to their planes, rotating with angular speeds $10\text{ rad/s}$ & $4\text{ rad/s}$ respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is _________ J.

Consider a Disc of mass $5\text{ kg}$, radius $2\text{ m}$, rotating with angular velocity of $10\text{ rad/s}$ about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is ________ J.

A uniform rod $AB$ of mass $2\text{ kg}$ and length $30\text{ cm}$ is at rest on a smooth horizontal surface. An impulse of force $0.2\text{ N s}$ is applied to end B. The time taken by the rod to turn through at right angles will be $\frac{\pi}{x}\text{ s}$, where $x =$ _______.

A mass '$M$' is moving with constant velocity parallel to $\mathrm{X}$-axis. Its angular momentum with respect to the origin is

What is the moment of inertia of the electron moving in second Bohr orbit of hydrogen atom? [ $\mathrm{h} =$ Planck's constant, $\mathrm{m} =$ mass of electron, $\varepsilon_0 =$ permittivity of free space, $\mathrm{e} =$ charge on electron]

A particle of mass '$\mathrm{m}$' is rotating along a circular path of radius '$r$' having angular momentum '$L$'. The centripetal force acting on the particle is given by

A particle of mass 100 g is projected at time t = 0 with a speed 20 ms$-1$ at an angle 45$^\circ$ to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time t = 2s is found to be $\sqrt{\mathrm{K}}\text{ kg m}^{2}\text{/s}$. The value of K is ___________. (Take g = 10 ms$-2$)

A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity $\omega$. A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If E be the initial Kinetic energy of the system, then final Kinetic energy will be $\frac{E}{x}$. The value of $x$ is

If the earth suddenly shrinks to $\frac{1}{64}$ th of its original volume with its mass remaining the same, the period of rotation of earth becomes $\frac{24}{x}$h. The value of x is __________.

A thin uniform circular disc of mass '$M$' and radius '$R$' is rotating with angular velocity '$\omega$', in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same radius but of mass $\left(\frac{M}{2}\right)$ is placed gently on the first disc co-axially. The new angular velocity will be

The position vector of 1 kg object is$\vec{r} = \left(3\hat{i} - \hat{j}\right)m$ and its velocity$\vec{v} = \left(3\hat{j} + \hat{k}\right)ms^{-1}$. The magnitude of its angular momentum is $\sqrt{x}$ Nm where x is ___________.

A spherical shell of 1 kg mass and radius R is rolling with angular speed $\omega$ on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin O is $\frac{a}{3}$ R2$\omega$. The value of a will be:

A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 rads${}^{-1}$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in rads${}^{-1}$).

A particle with position vector$\vec{r}$has a linear momentum $\vec{P}$. Which one of the following statements is true in respect of its angular momentum 'L' about the origin?

Angular momentum of a single particle moving with constant speed along circular path:

A body of mass '$\mathrm{m}$' and radius of gyration '$\mathrm{K}$' has an angular momentum $\mathrm{L}$. Its angular velocity is

A body of mass 'm' and radius of gyration 'K' has an angular momentum 'L'. Then its angular velocity is

A mass M hangs on a massless rod of length l which rotates at a constant angular frequency. The mass M moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular momentum of M about point A is LA which lies in the positive z direction and the angular momentum of M about point B is LB. The correct statement for this system is:

A particle of mass 'm' is moving in time 't' on a trajectory given by

$\vec{r} = 10\alpha t^{2}\hat{i} + 5\beta(t - 5)\hat{j}$

Where $\alpha$ and $\beta$ are dimensional constants.

The angular momentum of the particle becomes the same as it was for t = 0 at time t = ____________ seconds.

A child is standing with folded hands at the centre of the platform rotating about its central axis. The kinetic energy of the system is '$K$'. The child now stretches his arms so that the moment of inertia of the system becomes double. The kinetic energy of the system now is

Two discs have moments of inertia I1 and I2 about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $\omega$1 and $\omega$2 respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

A thin circular ring of mass M and radius r is rotating about its axis with an angular speed $\omega$. Two particles having mass m each are now attached at diametrically opposite points. The angular speed of the ring will become:

A wheel is rotating freely with an angular speed $\omega$ on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is:

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are 0.1 kg-m2 and 10 rad s–1 respectively while those for the second one are 0.2 kg-m2 and 5 rad s–1 respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is:

A circular disc of mass M and radius R is rotating about its axis with angular speed $\omega_1$. If another stationary disc having radius $\frac{R}{2}$ and same mass M is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed $\omega_2$ the energy lost in the process is p% of the initial energy. Value of p is __________.

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre _________.

Consider a uniform rod of mass M = 4m and length $\ell$ pivoted about its centre. A mass m moving with velocity v making angle $\theta = \frac{\pi}{4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is:

A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass 0.1 kg moving in a straight line with velocity 80 m/s hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad/s) of the rod immediately after the collision will be _________.

The time dependence of the position of a particle of mass m = 2 is given by$\vec{r}(t) = 2t\hat{i} - 3t^{2}\hat{j}$. Its angular momentum, with respect to the origin, at time t = 2 is

A thin smooth rod of length L and mass M is rotating freely with angular speed $\omega$0 about an axis perpendicular to the rod and passing through its center. Two beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be :-

If the angular momentum of a planet of mass m, moving around the Sun in a circular orbit is L, about the center of the Sun, its areal velocity is:

Two coaxial discs, having moments of inertia I1 and I1/2, are rotating with respective angular velocities $\omega$1 and $\omega$1/2, about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If Ef and Ei are the final and initial total energies, then (Ef - Ei) is:

Two particles A, B are moving on two concentric circles of radii R1 and R2 with equal angular speed $\omega$. At t = 0, their positions and direction of motion are shown in the figure.:

The relative velocity $\vec{V}_A - \vec{V}_B$ at t = $\frac{\pi}{2\omega}$ is given by:

A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude $\theta$0. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his center of mass moves by a distance $\ell$($\ell$ << L), is close to;

A uniform rod of length ' 6 L ' and mass ' 8 m ' is pivoted at its centre ' $C$ '. Two masses ' $m$ ' and ' $2m$ with speed $2v, v$ as shown strikes the rod and stick to the rod. Initially the rod is at rest. Due to impact, if it rotates with angular velocity ' $\omega$ ' then ' $\omega$ ' will be

A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere?

A thin uniform bar of length $L$ and mass $8$ m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2$v$ and $v$ respectively. The masses stick to the bar after collision at a distance $\frac{L}{3}$ and $\frac{L}{6}$ respectively from the center of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be:

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities $\omega_1$ and $\omega_2$. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is

A particle of mass m is moving along the side of a square of side ‘a’, with a uniform speed v in the x- y plane as shown in the figure:

Which of the following statements is false for the angular momentum $\vec{L}$ about the origin?

A cubical block of side 30 cm is moving with velocity 2 ms−1 on a smooth horizontal surface. The surface has a bump at a point O as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is:

An automobile moves on a road with a speed of 54 km h${}^{-1}$. The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kg m2. If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is

A force $\vec{F} = \alpha\hat{i} + 3\hat{j} + 6\hat{k}$ is acting at a point$\vec{r} = 2\hat{i} - 6\hat{j} - 12\hat{k}$. The value of $\alpha$ for which angular momentum about origin is conserved is

A mass m moves in a circle on a smooth horizontal plane with velocity v0 at a radius R0. The mass is attached to a string which passes through a smooth hole in the plane as shown. the tension in the string is increased gradually and finally m moves in a circle of radius $\frac{R_0}{2}$.

The final value of the kinetic energy is

A ball of mass 160 g is thrown up at an angle of 60° to the horizontal at a speed of 10 ms${}^{-1}$. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly (g = 10 ms${}^{-2}$)

A particle is moving in a circular path of radius a, with a constant velocity $v$ as shown in the figure. The center of circle is marked by 'C'. The angular momentum from the origin O can be written as:

A bob of mass $m$ attached to an inextensible string of length $l$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega\, rad/s$ about the vertical. About the point of suspension:

A particle of mass 2 kg is moving such that at time t, its position, in meter, is given by $\vec{r}(t) = 5\hat{i} - 2t^{2}\hat{j}$. The angular momentum of the particle at $t = 2s$ about the origin in kg m$2$ s$-1$ is:

Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D1 has 2 kg mass and 0.2 m radius and initial angular velocity of 50 rad s${}^{-1}$. Disc D2 has 4 kg mass, 0.1 m radius and initial angular velocity of 200 rad s${}^{-1}$. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad s${}^{-1}$) of the system is

A ring of mass M and radius R is rotating about its axis with angular velocity $\omega$. Two identical bodies each of mass m are now gently attached at the two ends of a diameter of the ring. Because of this, the kinetic energy loss will be:

A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds into it will be:

When a mass is rotating in a plane about a fixed point, its angular momentum is directed along

A circular platform is mounted on a frictionless vertical axle. Its radius R = 2 m and its moment of inertia about the axle is 200 kg m2. It is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 ms${}^{-1}$ relative to the ground. Time taken by the man to complete one revolution is

A small mass attached to a string rotates on a frictionless table top as shown. If the tension in the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of 2, the kinetic energy of the mass will

A small particle of mass $m$ is projected at an angle $\theta$ with the $x$-axis with an initial velocity $v_0$ in the $x\text{-}y$ plane as shown in the figure. At a time $t < \frac{v_0\sin\theta}{g},$ the angular momentum of the particle is ................,

where $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along $x, y$ and $z$-axis respectively.