System of Particles and Rotational Motion

Which physical quantity remains constant for a system of particles if no external force acts on it?

A uniform wire is bent into the shape of a circle. Its center of mass is at:

If the center of mass of a system is at the origin, the sum of the products of the mass and the x-coordinate for all particles is:

In a two-particle system, if the masses are $m$ and $3m$, the ratio of the distance of the center of mass from $m$ to its distance from $3m$ is:

A uniform wire of length $L$ is bent into a 'V' shape with an angle of $60^\circ$. The distance of the center of mass from the vertex is:

A bomb projected with a certain velocity at an angle explodes in mid-air. The center of mass of the fragments moves:

Two balls of masses $M$ and $2M$ are thrown simultaneously. $M$ is thrown vertically up with $20\text{ m/s}$ and $2M$ is thrown at an angle of $30^\circ$ with the horizontal with $20\text{ m/s}$. The acceleration of their center of mass is:

A uniform solid right circular cone of base radius $R$ and height $h$ is joined to a solid cylinder of same radius and height $h$. The center of mass of the composite body from the base of the cylinder (opposite to the cone) is:

A thin rod of length $L$ is placed along the $x$-axis from $x=0$ to $x=L$. Its linear density is $\lambda = ax^2 + b$. The $x$-coordinate of its center of mass is:

Two particles move towards each other under mutual gravitational force. If at some instant their velocities are $v$ and $2v$, the velocity of their center of mass is:

For a rigid body, the direction of angular momentum is always along the:

Which of the following describes the relationship between angular momentum and linear momentum for a particle?

If the angular momentum of a body increases at a constant rate, then the torque acting on it must be:

A steering wheel is turned using two equal and opposite forces. These forces constitute a:

The torque acting on a body is the rotational analogue of:

A particle performs uniform circular motion. The torque on the particle about the center of the circle is:

For a rigid body rotating about a fixed axis, the angular momentum vector $\vec{L}$ and the angular velocity vector $\vec{\omega}$:

A mass $m$ attached to a string of length $l$ is rotated in a horizontal circle on a frictionless table. The other end of the string is pulled through a hole in the table so that the radius of the circle reduces to $l/2$. The initial kinetic energy was $K$. The new kinetic energy will be:

An impulsive force $F$ acts on a sphere of mass $M$ and radius $R$ at a height $h$ above the center of the sphere. The initial angular velocity acquired by the sphere is:

A rigid body is rotating about an axis. To stop the rotation, we have to apply:

The moment of a force depends on:

A person carrying a bucket of water in one hand leans to the other side to:

The magnitude of torque is maximum when the angle between $\vec{r}$ and $\vec{F}$ is:

For a body in unstable equilibrium, a small displacement causes its center of gravity to:

Two forces of $10\, N$ each act on a rod of length $2\, m$ at its ends in opposite directions, making an angle of $30^\circ$ with the rod. The magnitude of the torque of this couple is:

A uniform rod of length $1\, m$ is balanced at its center. If $2\, kg$ is hung at $20\, cm$ and $3\, kg$ is hung at $30\, cm$ from one end, where should a $5\, kg$ mass be hung from the same end to maintain equilibrium?

A rigid body is acted upon by three non-parallel forces. If the body is in equilibrium, the lines of action of these forces must:

A square frame of four uniform rods each of mass $M$ and length $L$ is hung from a corner. A mass $2M$ is attached to the adjacent corner. The angle the diagonal through the point of suspension makes with the vertical is:

A uniform disc of mass $M$ and radius $R$ is leaning against a step of height $R/4$. A force $F$ is applied at the center. The minimum $\mu$ at the floor for the disc to climb without slipping at the floor is:

A solid sphere of mass $M$ is held in equilibrium on a rough incline by a horizontal string attached to its top. If the incline angle is $\theta$, the tension in the string is:

The moment of inertia of a thin uniform semicircular wire of mass $M$ and radius $R$ about an axis passing through its center (of the circle) and perpendicular to its plane is:

What happens to the moment of inertia of a rotating disc if its temperature is increased? (Assume it expands)

Two point masses each of mass $m$ are placed at a distance $r$ from each other. The moment of inertia of the system about an axis passing through their center of mass and perpendicular to the line joining them is:

The moment of inertia of a rectangular plate of mass $M$, length $l$ and breadth $b$ about an axis passing through its center and perpendicular to its plane is:

A square frame is constructed using four identical uniform thin rods, each of mass $m$ and length $L$. What is the moment of inertia of this frame about an axis passing through one of the corners and perpendicular to the plane of the frame?

The moment of inertia of a uniform square plate of side $a$ and mass $M$ about one of its diagonals is:

If the distance between two parallel axes is $d$, the moment of inertia of a body about the second axis $I$ is related to $I_{cm}$ by $I = I_{cm} + Md^2$. Here $I_{cm}$ must be the moment of inertia about:

A uniform rectangular plate has mass $M$ and sides $L$ and $2L$. Its moment of inertia about an axis passing through one of the corners and perpendicular to the plate is:

Three identical solid cylinders, each of mass $M$ and radius $R$, are placed on a horizontal surface such that their axes are parallel and they touch each other. The moment of inertia of the system about an axis passing through the contact point of two cylinders and parallel to their axes is:

A thin rod of mass $M$ and length $L$ is bent into a 'U' shape with three equal segments. The moment of inertia of this shape about an axis passing through the midpoints of the two parallel arms is:

For a system of particles, if the external force is zero, then $\frac{dP}{dt} = 0$, where $P$ is:

Two bodies of masses $2\text{ kg}$ and $3\text{ kg}$ are at $(0,0)$ and $(5,0)$ respectively. The $x$-coordinate of the center of mass is:

The center of mass of a circular disc of uniform thickness and uniform density lies at:

If a man walks on a boat in still water, the center of mass of the system remains stationary because:

Consider a system of two particles of masses $m_1$ and $m_2$. If $m_1$ is pushed towards the centre of mass through a distance $d$, by what distance should $m_2$ be moved so as to keep the centre of mass at the same position?

An explosion breaks a rock into three parts in a horizontal plane. Two of them go off at right angles to each other. The first part of mass $1\text{ kg}$ moves with a speed of $12\text{ m/s}$ and the second part of mass $2\text{ kg}$ moves with $8\text{ m/s}$ speed. If the third part flies off with $4\text{ m/s}$ speed, its mass is:

A $3\text{ kg}$ particle is at $(0,0)$ and a $1\text{ kg}$ particle is at $(4,0)$. If the $3\text{ kg}$ particle is moved to $(1,0)$, the centre of mass moves by:

Two particles of masses $100$ g and $300$ g at a given time have positions $2\hat{i}+5\hat{j}+13\hat{k}$ and $-6\hat{i}+4\hat{j}-2\hat{k}$ m respectively and velocities $10\hat{i}-7\hat{j}-3\hat{k}$ and $7\hat{i}-9\hat{j}+6\hat{k}$ m/s respectively. The velocity of the center of mass is:

A system of two particles is moving with some velocity. If an internal force acts between them, which of the following remains constant?

A uniform square plate has a mass of $M$ and side $a$. If one-fourth of the plate (a square of side $a/2$) is removed from one corner, the distance of the center of mass of the remaining part from the center of the original plate is:

Which of the following does NOT affect the total linear momentum of a system?

Two particles $A$ and $B$ of equal mass are moving with same speed $v$ but in opposite directions. The total momentum of the system is:

Internal forces cancel out in the sum $\sum F_i$ because of:

Rocket propulsion is based on the principle of:

A system consists of two particles with masses $m_1 = 2\text{ kg}$ and $m_2 = 3\text{ kg}$. Their initial velocities are $\vec{v}_1 = 5\hat{i}\text{ m/s}$ and $\vec{v}_2 = 10\hat{j}\text{ m/s}$ respectively. If a constant external force $\vec{F}_{ext} = 10\hat{i}\text{ N}$ acts on the system for a time interval of $3\text{ s}$, what is the magnitude of the final total linear momentum of the system?

A constant force $F$ acts on a system of particles for a time interval $\Delta t$. The change in the total linear momentum of the system is:

A nucleus at rest disintegrates into two nuclear fragments with velocities in the ratio 2:1. The ratio of their nuclear sizes (radii) is:

A rocket burns fuel at a rate of $50 \text{ kg/s}$. The exhaust gases are ejected at a speed of $4000 \text{ m/s}$ relative to the rocket. The thrust on the rocket is:

A system of $2$ particles has $m_1=2 \text{ kg}$ and $m_2=3 \text{ kg}$. If $m_1$ has acceleration $a_1 = 2\hat{i} + 3\hat{j}$ and $m_2$ has acceleration $a_2 = -1\hat{i} - 2\hat{j}$, the acceleration of the center of mass is:

A block of mass $m$ moving with speed $v$ collides with another block of mass $2m$ at rest. If the collision is head-on and perfectly elastic, the fractional loss of energy of the first block is:

If $\vec{A}$ is along the x-axis and $\vec{B}$ is along the negative y-axis, then $\vec{A} \times \vec{B}$ is along:

If $\vec{A} \times \vec{B} = \vec{C}$, then $\vec{C}$ is:

The magnitude of the vector product of two vectors is equal to the area of the:

If the angle between $\vec{A}$ and $\vec{B}$ is $180^\circ$, then $\vec{A} \times \vec{B}$ is:

Given a vector $\vec{A} = 3\hat{i} + 4\hat{j}$ which lies in the XY plane, find the result of the vector product $\vec{A} \times \hat{k}$, where $\hat{k}$ is the unit vector along the Z-axis.

Calculate the Y-component (j-component) of the vector product $\vec{u} \times \vec{v}$, given that $\vec{u} = 2\hat{i} - \hat{j}$ and $\vec{v} = \hat{j} + 3\hat{k}$.

In a rotational system, the linear velocity of a particle is given by the vector product $\vec{v} = \vec{\omega} \times \vec{r}$. If at a specific moment, the position vector $\vec{r}$ is directed along the positive X-axis and the linear velocity $\vec{v}$ is directed along the positive Y-axis, what is the direction of the angular velocity vector $\vec{\omega}$?

Three non-zero vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ satisfy the specific geometric relations $\vec{A} \times \vec{B} = \vec{C}$ and $\vec{B} \times \vec{C} = \vec{A}$. Based on these properties, what is the magnitude of vector $\vec{B}$ and the relative orientation of the three vectors?

If the area of a parallelogram with sides $\vec{A}$ and $\vec{B}$ is $20$ units, what is the area of a triangle with the same sides?

The cross product of $\vec{r} = 2\hat{i} + 3\hat{j}$ and $\vec{F} = 5\hat{i} - 2\hat{j}$ results in a torque vector along:

If the radius of a circular path is constant, the linear velocity is ______ to the angular velocity.

The time taken to complete one full rotation is called:

For a rotating body, the linear velocity is maximum for a particle located:

Angular velocity is an example of an:

A rigid body rotates about a fixed axis with a constant angular velocity $\vec{\omega}$. Let $\vec{r}$ be the position vector of a particle in the body relative to an origin $O$ located on the axis of rotation. If $\theta$ is the angle between the angular velocity vector $\vec{\omega}$ and the position vector $\vec{r}$, which of the following expressions correctly represents the magnitude of the linear velocity $v$ of the particle?

In the relation $\vec{v} = \vec{\omega} \times \vec{r}$, if $\vec{r}$ is the position vector from the origin on the axis of rotation, then $\vec{v}$ is:

A particle moves on a circle of radius $r$. If its angular displacement is $\theta = 2t^2 + 5$, its angular velocity at $t = 2$ s is:

A rigid body is rotating about an axis. If the distance of a point from the axis is doubled, its angular velocity will:

What is the angular velocity of the minute hand of a clock in rad/s?

A fly is sitting on a record player at a distance $r$ from the center. If the record player starts from rest and reaches an angular velocity $\omega$ in time $t$ with constant angular acceleration, the distance traveled by the fly is:

If the frequency of rotation is $f$, then the angular velocity is:

A wheel of radius $0.5 \ m$ rotates with angular acceleration of $2 \ rad/s^2$. The tangential acceleration of a point on the rim is:

Angular acceleration is parallel to angular velocity only when:

Which quantity is the rotational analogue of mass?

A flywheel starts from rest and undergoes a two-stage rotational motion as described in the flowchart. In Phase 1, it rotates with a constant angular acceleration of $2\, rad/s^2$ for $4\, s$. In Phase 2, it continues to rotate for another $3\, s$ at the constant angular velocity it reached at the end of Phase 1. What is the total angular displacement of the flywheel at the end of $7\, s$?

A body performing uniform circular motion has:

The angular speed of the second hand of a watch is:

The time period of a satellite in a circular orbit around the Earth is $T$. Its angular velocity is:

A solid cylinder rolls down an inclined plane of height $h$. The velocity of the center of mass when it reaches the bottom is:

If the angular velocity of a particle is $\vec{\omega} = \hat{i} + 2\hat{j} + 3\hat{k}$ and the position vector is $\vec{r} = 2\hat{i} - \hat{j} + \hat{k}$, then the linear velocity is:

Angular momentum is the moment of:

For a ring of mass $M$ and radius $R$, the moment of inertia about its diameter is:

Work done by a torque is maximum when the angle of rotation is:

The relation between linear acceleration ($a$) and angular acceleration ($\alpha$) for a point at distance $r$ is:

A uniform circular disc of mass $2$ kg and radius $20$ cm is free to rotate about a fixed axle passing through its center. Two tangential forces $F_1 = 10$ N and $F_2 = 6$ N are applied to the rim such that they tend to rotate the disc in opposite directions (one clockwise and one counter-clockwise). What is the resulting angular acceleration of the disc?

What is the torque of the force $\vec{F} = (2\hat{i} - 3\hat{j} + 4\hat{k})$ N acting at the point $\vec{r} = (3\hat{i} + 2\hat{j} + 3\hat{k})$ m about the origin?

The moment of inertia of a square plate of side $a$ and mass $M$ about an axis passing through its center and perpendicular to its plane is:

A uniform disc of mass $M$ and radius $R$ is rolling without slipping on a horizontal surface. What is the total kinetic energy of the disc if the velocity of its center of mass is $v$?

A point mass $m$ is moving in the $xy$-plane with a velocity $v = v_x i + v_y j$. Its position vector is $r = x i + y j$. The $z$-component of its angular momentum about the origin is:

A solid sphere and a solid cylinder have the same mass and radius. They are rolling on a horizontal surface with the same kinetic energy. The ratio of their translational velocities $v_{sphere} : v_{cylinder}$ is:

A ring and a disc of the same mass and radius rotate about their central axes with the same angular velocity. Which has higher angular momentum?

Which of the following quantities has the same dimensions as Planck's constant?