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 $$\omega$$, 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 it 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 axis;

Case (b) : The disc with its face making an angle of 45$$^\circ$$ with xy-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 $$\omega$$ about the z-axis.

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

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 xy-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 $${\overrightarrow L _O}$$ and $${\overrightarrow L _P}$$, respectively, then

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 (|$$\tau$$|) about O, as a function of time is best represented by which plot ?

A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant $$k$$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity $${\overrightarrow V _0} = {V_0}\widehat i$$. the coefficient of friction is $$\mu$$.

The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is :