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

A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as \left( {{{{\sigma _0}} \over r}} \right) , then the radius of gyration of the disc about its axis passing through the centre is:

A uniform rod of length $$\ell$$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is T(x) at a distance x from the axis, then which of the following graphs depicts it most closely?

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 particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A, as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. when the particle reaches point b, its angular momentum about O will be : (Take g = 10 m/s2)

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 $${\overrightarrow{V}}_A-{\overrightarrow{V}}_B$$ at t = {\pi \over {2\omega }} is given by :

A thin circular plate of mass M and radius R has its density varying as $$\rho$$(r) = $$\rho$$0r with $$\rho$$0 as constant and r is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = aMR2. The value of the coefficient a is :

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 :

A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of 30o from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad s$$-$$1) will be (g = 10 ms$$-$$2)

To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is $$\mu$$, the torque, applied by the machine on the mop is -

A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is -

A rigid massless rod of length 3l has two masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be -

Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is :

A slab is subjected to two forces $$\overrightarrow{F_1}$$ and $$\overrightarrow{F_2}$$ of same magnitude F as shown in the figure. Force $$\overrightarrow{F_2}$$ is in XY-plane while force $$\overrightarrow{F_1}$$ acts along z = axis at the point $$\left(2\overrightarrow{i}+3\overrightarrow{j}\right).$$. The moment of these forces about point O will be :

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5m, the angle between the force and the position vector is (in radians) :

A circular disc D1 of mass M and radius R has two identical discs D2 and D3 of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO' ,passing through the centre of D1 as shown in the figure, will be :

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) :

Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is :

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is $$\theta$$, thrown :

The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is 'I(x)'. Which one of the graphs represents the variation of I(x) with x correctly ?