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**Previous Years JEE Advanced Questions**

**Paragraph for Questions 10 and 11**

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed , the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the z-axis, and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points P and Q). Both these motions have the same angular speed $\omega$ in this case.

Now consider two similar systems as shown in the figure : case (A) the disc with its face vertical and parallel to x-z plane; Case (B) the disc with its face making an angle of $45^{\circ}$ with x-y plane and its horizontal diameter parallel to x-axis. In both the cases, the disc is welded at point P, and the systems are rotated with constant angular speed about the z-axis.

**Paragraph 1**

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity is an example of a non-inertial frame of reference. The relationship between the force $\overrightarrow{\mathrm{F}}_{\mathrm{rot}}$ experienced by a particle of mass m moving on the rotating disc and the force $\overrightarrow{\mathrm{F}}_{\mathrm{in}}$ experienced by the particle in an inertial frame of reference is

$\overrightarrow{\mathrm{F}}_{\mathrm{rot}}=\overrightarrow{\mathrm{F}}_{\mathrm{in}}+2 \mathrm{m}\left(\overrightarrow{\mathrm{v}}_{\mathrm{rot}} \times \vec{\omega}\right)+\mathrm{m}(\vec{\omega} \times \overrightarrow{\mathrm{r}}) \times \vec{\omega}$

where $\vec{v}_{\text {rot }}$ is the velocity of the particle in the rotating frame of reference and $\overrightarrow{\mathrm{r}}$ is the position

vector of the particle with respect to the centre of the disc.

Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed about its vertical axis through its center. We assign a coordinate system with the origin at the centre of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis $(\vec{\omega}=\omega \hat{k})$ A small block of mass m is gently placed in the slot at $\overrightarrow{\mathrm{r}}=(\mathrm{R} / 2) \hat{\mathrm{i}}$ at $\mathrm{t}=0$ and is constrained to move only along the slot.

**Paragraph for Questions 27 and 28**

One twirls a circular ring (of mass M and radius R) near the tip of one’s finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity $\omega_{0}$. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is µ and the acceleration due to gravity is g.