Three identical spherical shells, each of mass $m$ and radius $r$ are placed as shown in the figure. Consider an axis $XX^{\prime}$ which is touching to two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about$XX^{\prime}$axis is

A body of mass 1 kg begins to move under the action of a time-dependent force F = (2t î + 3t² ĵ) N. What power will be developed by the force at the time t?

From a disc of radius $R$ and mass $M,$ a circular hole of diameter $R,$ whose rim passes through the centre, is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis passing through the centre?

A uniform circular disc of radius $50$ cm at rest is free to turn about an axis which is perpendicular to its plane and passes through its centre. It is subjected to a torque which produces a constant angular acceleration of $2.0$ rads$^{-2}.$ Its net acceleration in $$ms$^{-2}$ at the end of $2.0$ s is approximately

Two astronauts are floating in gravitational free space after having lost contact with their spaceship. The two will

One end of a string of length $l$ is connected to a particle of mass $m$ and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $v,$ the net force on the particle (directed towards center) will be $(T$ represents the tension in the string$)$

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

Two discs of the same moment of inertia rotating about their regular axis passing through the center and perpendicular to the plane of the 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

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

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

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 mass $m$ is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

A solid cylinder of mass $2$ kg and radius $4$ cm is rotating about its axis at the rate of $3$ rpm. The torque required to stop it after $2\pi$ revolutions is:

A particle moving with velocity $V$ is acted by three forces shown by the vector triangle PQR. The velocity of the particle will:

Two particles A and B are moving in uniform circular motion in concentric circles of radii $r_A$ and $r_B$ with speed $v_A$ and $v_B$ respectively. Their time period of rotation is the same. The ratio of angular speed of A to that of B will be:

A block of mass $10$ kg is in contact against the inner wall of a hollow cylindrical drum of radius $1$ m. The coefficient of friction between the block and the inner wall of the cylinder is $0.1.$ The minimum angular velocity needed for the cylinder to keep the block stationary when the cylinder is vertical and rotating about its axis, will be: $(g = 10$ m/s$^2)$

A disc of radius $2$ m and mass $100$ kg rolls on a horizontal floor. Its centre of mass has speed of $20$ cm/s. How much work is needed to stop it?

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 center 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 of $3\hat{j}$ N acts on a particle whose position vector is $2\hat{k}$ m.

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 and another unknown mass $m$ is suspended from the rod at $160$ cm mark.$\newline$Find the value of $m$ such that the rod is in equilibrium.$\newline (g = 10$ m/s$^2)$

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 center of the ring and perpendicular to the plane of the ring is $K$ times $MR^2.$ Then the value of $K$ is:

A shell of mass $m$ is at rest initially. It explodes into three fragments having mass in the ratio $2 : 2: 1.$ If the fragments having equal mass fly off along mutually perpendicular directions with speed $v,$ the speed of the third (lighter) fragment 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 center of mass of the system from the $10$ kg mass 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:

The angular speed of a flywheel moving with uniform angular acceleration changes from $1200$ rpm to $3120$ rpm in $16$ seconds. The angular acceleration in rad/s$^2$ is:

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

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:

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 $2400$ g cm$^2.$ The length of the 400 g rod is nearly: