Rotational Motion

A bob of mass m attached to an inextensible string of length $l$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega$ rad/s about the vertical. About the point of suspension :

A cylinder of mass $M_c$ and sphere of mass $M_s$ are placed at points A and B of two inclines, respectively. (See Figure). If they roll on the incline without slipping such that their accelerations are the same, then the ratio $\dfrac{\sin\theta_c}{\sin\theta_s}$ is :

A particle is moving in a circular path of radius a, with a constant velocity $v$ as shown in the figure. The center of circle is marked by 'C'. The angular momentum from the origin O can be written as :

Consider a cylinder of mass M resting on a rough horizontal rug that is pulled out from under it with acceleration 'a' perpendicular to the axis of the cylinder. What is $F_{\text{friction}}$ at point P ? It is assumed that the cylinder does not slip.

A ball of mass 160 g is thrown up at an angle of 60° to the horizontal at a speed of 10 ms$^{-1}$. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly ($g=10$ ms$^{-2}$)

A uniform circular disc of mass 50 kg and radius 0.4 m is rotating with an angular velocity of 10 rad s$^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass 6.25 kg and radius 0.2 m, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in rad s$^{-1}$) of the system is

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is

A thin uniform rod, pivoted at $O$, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At time $t=0$, a small insect starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reaches the end of the rod at $t=T$ and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $(|\vec\tau|)$ on the system about $O$, as a function of time is best represented by which plot?

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x$-$y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by $\vec L_O$ and $\vec L_P$ respectively, then

A lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density and radius $2R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_O$ and $I_P$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\dfrac{I_P}{I_O}$ to the nearest integer is

Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R$. The velocity of projection is in the $y$-$z$ plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed $\frac18$ rotation, (ii) their range is less than half the disc radius, and (iii) $\omega$ remains constant throughout. Then

Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

Two solid cylinders $P$ and $Q$ of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder $P$ has most of its mass concentrated near its surface, while $Q$ has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

The figure shows a system consisting of (i) a ring of outer radius $3R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius $2R$ rotating anti-clockwise with angular speed $\omega/2$. The ring and disc are separated by frictionless ball bearings. The system is in the $x$-$z$ plane. The point $P$ on the inner disc is at a distance $R$ from the origin, where $OP$ makes an angle of 30° with the horizontal. Then with respect to the horizontal surface,

A metal rod of length 'L' and mass 'm' is pivoted at one end. A thin disk of mass 'M' and radius 'R' (< L) is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of 0.3 m/s². The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is (P/10). The value of P is

Four solid spheres each of diameter $\sqrt5$ cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is $N\times10^{-4}\ \mathrm{kg\,m^2}$, then $N$ is

A thin ring of mass 2 kg and radius 0.5 m is rolling without slipping on a horizontal plane with velocity 1 m/s. A small ball of mass 0.1 kg, moving with velocity 20 m/s in the opposite direction, hits the ring at a height of 0.75 m and goes vertically up with velocity 10 m/s. Immediately after the collision

A binary star consists of two stars A (mass $2.2M_s$) and B (mass $11M_s$), where $M_s$ is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is