System of Particles and Rotational Motion PYQs

The angular speed of a flywheel is increased from 600 rpm to 1200 rpm in 10 s. The number of revolutions completed by the flywheel during this time is:

A thin wire of length 'L' and linear mass density 'm' is bent into a circular ring (in x-y plane) with centre C as shown in figure. The moment of inertia of the ring about an axis yy′ will be:

A sphere of radius $R$ is cut from a larger solid sphere of radius $2R$ as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the $Y$-axis is:

A wheel of a bullock cart is rolling on a level road as shown in the figure below. If its linear speed is v in the direction shown, which one of the following options is correct (P and Q are any highest and lowest points on the wheel, respectively)?

The moment of inertia of a thin rod about an axis passing through its mid point and perpendicular to the rod is $2400\,\text{g cm}^2$. The length of the $400\,\text{g}$ rod is nearly:

The ratio of radius of gyration of a solid sphere of mass M and radius R about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is :

The angular acceleration of a body, moving along the circumference of a circle, is :

The ratio of the radius of gyration of a thin uniform disc about an axis passing through its centre and normal to its plane to the radius of gyration of the disc about its diameter is

Two objects of mass 10 kg and 20 kg respectively are connected to the two ends of a rigid rod of length 10 m with negligible mass. The distance of the centre of mass of the system from the 10 kg mass is:

The angular speed of a fly wheel moving with uniform angular acceleration changes from 1200 rpm to 3120 rpm in 16 seconds. The angular acceleration in rad/s² is:

From a circular ring of mass $M$ and radius $R$, an arc corresponding to a $90^\circ$ sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is $KMR^2$. The value of $K$ is:

A uniform rod of length 200 cm and mass 500 g is balanced on a wedge placed at 40 cm mark. A mass of 2 kg is suspended from the rod at 20 cm mark and another unknown mass m is suspended from the rod at 160 cm mark. Find the value of m such that the rod is in equilibrium. (g = 10 m/s²)

Two particles of mass 5 kg and 10 kg respectively are attached to the two ends of a rigid rod of length 1 m with negligible mass. The centre of mass of the system from the 5 kg particle is nearly at a distance of:

Find the torque about the origin when a force $3\hat{j}\,\text{N}$ acts on a particle whose position vector is $2\hat{k}\,\text{m}$.

A solid cylinder of mass $2\ \text{kg}$ and radius $50\ \text{cm}$ rolls up an inclined plane of angle $30^\circ$. The centre of mass of the cylinder has speed of $4\ \text{m/s}$. The distance travelled by the cylinder on the inclined surface will be, [take $g = 10\ \text{m/s}^2$]

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 solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy ($K_t$) as well as rotational kinetic energy ($K_r$) simultaneously. The ratio $K_t : (K_t + K_r)$ for the sphere is:

Three objects; A : (a solid sphere), B : (a thin circular disk) and C : (a circular ring), each have the same mass M and radius R. They all spin with the same angular speed $\omega$ about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation:

The moment of the force $\vec{F} = 4\hat{i} + 5\hat{j} - 6\hat{k}$ acting at point $(2,0,-3)$ about the point $(2,-2,-2)$ is given by:

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

A rope is wound around a hollow cylinder of mass $3\,\text{kg}$ and radius $40\,\text{cm}$. What is the angular acceleration of the cylinder if the rope is pulled with a force of $30\,\text{N}$?

Which of the following statements are correct? (a) Centre of mass of a body always coincides with the centre of gravity of the body. (b) Centre of mass of a body is the point at which the total gravitational torque on the body is zero. (c) A couple on a body produces both translational and rotational motion in a body. (d) Mechanical advantage greater than one means that small effort can be used to lift a large load.

Two rotating bodies $A$ and $B$ of masses $m$ and $2m$ with moments of inertia $I_A$ and $I_B$ ($I_B$ > $I_A$) have equal kinetic energy of rotation. If $L_A$ and $L_B$ are their angular momenta respectively, then:

A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $(E_{sphere}/E_{cylinder})$ will be:

A light rod of length $l$ has two masses $m_1$ and $m_2$ attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is:

An automobile moves on a road with a speed of $54\,\text{kmh}^{-1}$. The radius of its wheels is $0.45\,\text{m}$ and the moment of inertia of a wheel about its axis of rotation is $3\,\text{kg m}^2$. If the vehicle is brought to rest in $15\,\text{s}$, the magnitude of average torque transmitted by its brakes to a wheel is:

Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length $L$ and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point $P$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity $\omega_0$ is minimum, is given by:

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 the origin is conserved is:

Two stones of masses $m$ and $2m$ are whirled in horizontal circles, the heavier one in a radius $\dfrac{r}{2}$ and the lighter one in radius $r$. The tangential speed of the lighter stone is $n$ times that of the heavier stone when they experience the same centripetal force. The value of $n$ is:

On a frictionless surface, a block of mass $M$ moving at speed $v$ collides elastically with another block of the same mass $M$ which is initially at rest. After collision the first block moves at an angle $\theta$ to its initial direction and has a speed $\dfrac{v}{3}$. The second block's speed after the collision is:

A solid cylinder of mass $50\,\text{kg}$ and radius $0.5\,\text{m}$ is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and the other hanging freely. The tension in the string required to produce an angular acceleration of $2$ revolutions $\text{s}^{-2}$ is:

The ratio of the accelerations for a solid sphere (mass 'm' and radius 'R') rolling down an incline of angle $\theta$ without slipping and slipping down the incline without rolling is:

If a is the length of the side of a cube, the distance between the body centered atom and one corner of the cube will be:

A rod $PQ$ of mass $M$ and length $L$ is hinged at end $P$. The rod is kept horizontal by a massless string tied to point $Q$ as shown in the figure. When the string is cut, the initial angular acceleration of the rod is:

A small object of uniform density rolls up a curved surface with an initial velocity $v$'. It reaches up to a maximum height $\dfrac{3v^2}{4g}$ with respect to the initial position. The object is

The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through:

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

Three masses are placed on the x-axis: 300 g at origin, 500 g at x = 40 cm and 400 g at x = 70 cm. The distance of the centre of mass from the origin 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:

From a circular disc of radius $R$ and mass $9M$, a small disc of mass $M$ and radius $\dfrac{R}{3}$ is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is

A solid cylinder and a hollow cylinder, both of the same mass and same external diameter are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first?

Consider the following statements: (i) Centre of gravity (C.G.) of a body is the point at which the weight of the body acts. (ii) Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius. (iii) To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G. (iv) The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis. Which one of the following pairs of statements is correct?

A thin circular ring of mass M and radius r is rotating about its axis with constant angular velocity ω. Two objects each of mass m are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with angular velocity:

Two bodies of mass $1\,\text{kg}$ and $3\,\text{kg}$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The centre of mass of this system has a position vector:

Four identical thin rods each of mass $M$ and length $l$ form a square frame. The moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is:

A thin circular ring of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis perpendicular to its plane with angular velocity $\omega$. If two objects each of mass $m$ are attached gently to opposite ends of a diameter of the ring, then the ring will rotate with angular velocity:

If $\vec{F}$ is the force acting on a particle having position vector $\vec{r}$ and $\vec{\tau}$ be the torque of this force about the origin, then:

The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axis is

A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle between them is 90°. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is

A uniform rod is hinged at one end as shown in the figure. When the support is withdrawn, what is the acceleration of the center of mass immediately after release?

A pulley of radius 20 cm and moment of inertia 0.32 kg·m² is used to hang a 2 kg mass with a massless string. If the load is released from rest, calculate the acceleration of the block.

The moment of inertia of a uniform circular disc of radius R and mass M about an axis touching the disc at its diameter and normal to the disc is:

A tube of length L is filled completely with an incompressible liquid of mass M and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity ω. The force exerted by the liquid at the other end is:

A uniform rod of length l and mass m is free to rotate in a vertical plane about A. The rod is initially in horizontal position and is released. The initial angular acceleration of the rod is: (Moment of inertia of rod about A is (ml²/3))

A drum of radius $R$ and mass $M$ rolls down without slipping along an inclined plane of angle $\theta$. The frictional force:

Two bodies have moments of inertia I and 2I respectively about their axes of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio:

The moment of inertia of a uniform circular disc of radius $R$ and mass $M$ about an axis passing through the edge of the disc and normal to the disc is: