Motion of System of Particles and Rigid Body

For a system of particles, the rate of change of total linear momentum is equal to the:

The displacement of the centre of mass is given by $\Delta R_{cm} = \frac{\sum m_i \Delta r_i}{M}$. If a system is initially at rest and only internal forces act, then $\Delta R_{cm}$ is:

Two particles of mass $M$ and $2M$ are at a distance $D$ apart. The distance of the centre of mass from the particle of mass $M$ is:

The vector sum of all the internal forces in a system of particles is:

Two particles of masses $m_1$ and $m_2$ are at rest. They start moving towards each other under their mutual gravitational attraction. At any instant, the velocity of the centre of mass is:

If a particle of mass $m$ is removed from a system of total mass $M$ whose centre of mass was at the origin, the new centre of mass relative to the old one (if the removed particle was at $\vec{r}$) is:

A uniform right-angled triangular lamina has base $b$ and height $h$. The centre of mass is at a distance from the right-angled corner equal to:

A uniform solid cone of height $H$ and a uniform solid cylinder of height $H$ have the same base. If they are joined base-to-base, the distance of the centre of mass from the common base is:

Two particles of mass $m$ and $M$ are initially at rest at infinite distance. They move towards each other under mutual gravitational attraction. The velocity of the centre of mass at any instant is:

The centre of mass of a system of particles is at the origin. If a particle of mass $m$ at $(x, y, z)$ is removed, the new centre of mass of the remaining system of mass $M$ will be at:

If a body is rotating with constant angular velocity, the net torque acting on it is:

What is the resultant force of a couple consisting of two forces of $5\ N$ each?

The unit of torque in the SI system is:

An anticlockwise moment is produced when the force tends to rotate the body:

The moment of a force of $5\ N$ about a point $P$ is $2\ N \cdot m$. The perpendicular distance of the point $P$ from the line of action of the force is:

A force $\vec{F} = 3\hat{i} + 2\hat{j}$ acts on a particle located at $(x, y) = (2, -3)$. The torque about the origin is:

A particle of mass $m$ is moving with a constant velocity $\vec{v}$ along a line $y = c$. Its angular momentum about the origin is $L$. The torque acting on it about the origin is:

A uniform thin rod of total length $3L$ is bent into a U-shape consisting of three equal segments of length $L$. The shape is placed in the $xy$-plane such that the middle segment lies on the $x$-axis with its midpoint at the origin $O$. One arm of the U-shape extends from the point $(L/2, 0)$ to $(L/2, L)$. A force $\vec{F} = F\hat{k}$ is applied at the free end $(L/2, L)$ of this arm. What is the magnitude of the torque produced by this force about the origin?

When a diver pulls her hands and legs close to her body while spinning, the torque acting on her:

A force $\vec{F} = 4\hat{i} - 5\hat{j} + 3\hat{k}$ is acting at a point $\vec{r} = 2\hat{i} - 3\hat{j} - 2\hat{k}$. The torque about the point $(1, -1, 1)$ is:

If the radius of the circular path of a particle is doubled and its speed is halved, the angular momentum will:

When a ballet dancer pulls her arms inward, which quantity remains conserved if we ignore friction?

For a particle moving along a straight line not passing through the origin, the angular momentum about the origin:

In rotational motion, the quantity that plays the same role as mass in translational motion is used to calculate $L$. This quantity is:

A particle of mass $m$ is moving in a circular path of radius $r$ with a constant angular momentum $L$. If the radius of the circular path is reduced to $\frac{r}{3}$ without any external torque acting on the system, the new centripetal force required to maintain the motion will be:

Two small spheres, each of mass $m$, are mounted on a light rigid rod of length $2L$. The system rotates with an angular velocity $\omega$ about an axis perpendicular to the rod and passing through its center. If the spheres are moved from the ends of the rod to points at a distance of $L/2$ from the axis, what will be the new angular velocity of the system?

A thin circular ring of mass $M$ and radius $R$ is rotating about its axis with an angular speed $\omega$. Two particles, each of mass $m$, are attached gently to the opposite ends of a diameter of the ring. The new angular speed is:

A wooden rod of mass $M$ and length $L$ is pivoted at the top. A bullet of mass $m$ moving horizontally with speed $v$ hits the bottom of the rod and gets embedded. The angular momentum of the system about the pivot just after the collision is:

A particle of mass $m$ is moving with constant speed $v$ in a circle of radius $r$. If the radius is doubled while keeping the angular momentum constant, the new speed is:

A uniform disc of mass $M$ and radius $R$ is rotating about a horizontal axis through its center. A small piece of mass $m$ breaks off from the edge at the lowest point. Just after it breaks off, the angular momentum of the remaining disc about the center:

If the magnitude of the force applied to a wrench is doubled, what happens to the torque (assuming the distance remains the same)?

A wheelbarrow is an example of which class of lever?

A uniform meter scale is balanced at its $50 cm$ mark. If a weight is added to the $10 cm$ mark, the scale will tilt in which direction?

A body suspended from a point is in stable equilibrium when its center of gravity lies ________ the point of suspension.

A uniform rod is supported at its two ends by two strings. If the strings are not vertical but make equal angles with the horizontal, which of the following is true?

The center of mass of a system of three particles of masses $1\ g, 2\ g, 3\ g$ is at the point $(1, 1, 1)$. If a fourth particle of mass $4\ g$ is added at $(0, 0, 0)$, the new center of mass is at:

A man stands at one end of a $4\ m$ long boat of mass $40\ kg$ floating on still water. The man has a mass of $60\ kg$. If the man walks to the other end of the boat, how far does the boat move relative to the water?

A circular disc is rolling without slipping on a horizontal surface. Which point has the minimum instantaneous velocity?

A thin uniform rod of mass $M$ and length $L$ is rotating about a perpendicular axis through its center with angular velocity $\omega$. Two particles of mass $m$ are attached gently to the two ends of the rod. The new angular velocity is:

The moment of inertia of a uniform semicircular wire of mass $M$ and radius $R$ about a line perpendicular to the plane of the wire through its center of mass is:

A uniform square plate of side '$a$' has a moment of inertia $I_o$ about an axis perpendicular to its plane and passing through its center. Its moment of inertia about an axis in its plane passing through the center is:

Which of the following is NOT a unit of Moment of Inertia?

If the moment of inertia of a body is $4\text{ kg}\cdot\text{m}^2$ and its mass is $16\text{ kg}$, what is its radius of gyration?

Moment of inertia depends on the distribution of mass. For a fly-wheel, most of the mass is concentrated at:

A thin uniform rod of length $L$ and mass $M$ is bent into a complete circle. The moment of inertia of this circle about an axis passing through its center and perpendicular to its plane is:

Which of the following has the largest radius of gyration if all have the same radius $R$?

For a rolling solid sphere of mass $M$ and radius $R$, the ratio of rotational kinetic energy to total kinetic energy is:

A solid cylinder and a hollow cylinder of same mass and radius are released from the top of an inclined plane. Which one reaches the bottom first?

A thin uniform rod of length $L$ and mass $M$ is bent into a regular $n$-sided polygon. The moment of inertia about an axis through its center and perpendicular to its plane is:

Calculate the moment of inertia of a uniform disc of mass $M$ and radius $R$ about an axis passing through its center and making an angle of $45^\circ$ with its plane.

Find the moment of inertia of a thin rod of mass $M$ and length $L$ about an axis perpendicular to it and passing through a point $L/6$ from one end.

What is the ratio of $I_{tangent}$ to $I_{diameter}$ for a solid sphere?

Which of the following is true for any rigid body?

The moment of inertia of a hollow cylinder of mass $M$ and radius $R$ about its own axis (neglecting thickness) is:

The perpendicular axis theorem can be applied to find the moment of inertia of which of the following bodies?

A thin rod of mass $M$ and length $L$ has a radius of gyration $k$ about an axis passing through one end and perpendicular to the rod. The value of $k$ is:

A spherical shell of mass $M$ and radius $R$ has its moment of inertia calculated about an axis tangent to its surface. The value is:

An axis $XY$ passes through the center of a square lamina of side $L$ and is parallel to a diagonal. If the moment of inertia about this axis is $I$, find the moment of inertia about an axis parallel to $XY$ passing through a corner.

A uniform thin disc of mass $M$ and radius $R$ is cut into four equal quadrants. What is the moment of inertia of one quadrant about an axis passing through its vertex (the original center) and perpendicular to its plane?

A uniform thin square frame of side $a$ and mass $M$ is rotated about one of its diagonals. Calculate the moment of inertia using axis theorems.