$A$ thin hollow sphere of mass $m$ is completely filled with a liquid of mass $m$. When the sphere rolls with a velocity $v$, the kinetic energy of the system is (neglect friction):

$ABC$ is an equilateral triangle with $O$ as its centre. $\vec F_1, \vec F_2$ and $\vec F_3$ represent three forces acting along the sides $AB, BC$ and $AC$ respectively. If the total torque about $O$ is zero,then the magnitude of $\vec F_3$ is

$A$ thin uniform circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to the plane with angular velocity $\omega$. Another disc of same mass but half the radius is gently placed over it coaxially. The angular speed of the composite disc will be :

$A$ solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $M/8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2R$. If $I_1$ is the moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2/I_1 = . . . . . . $.

$A$ metal rod of length $L$ and mass $m$ is pivoted at one end. $A$ thin disk of mass $M$ and radius $R$ $(R < L)$ 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 a vertical plane after being released from the same displaced position. Which of the following statement$(s)$ is (are) true?

$A$ particle of mass $m$ is moving along the side of a square of side '$a$',with a uniform speed $v$ in the $x-y$ plane as shown in the figure. Which of the following statements is false for the angular momentum $\vec L$ about the origin?