$A$ yo-yo is resting on a rough horizontal table. Forces $F_1, F_2$ and $F_3$ are applied separately as shown. The correct statement is

$A$ homogeneous disc of mass $2 \ kg$ and radius $15 \ cm$ is rotating about its axis (which is fixed) with an angular velocity $4 \ rad/s$. The linear momentum of the disc is

$A$ solid sphere rotates about a vertical axis on a frictionless bearing. $A$ massless cord passes around the equator of the sphere,then passes over a solid cylinder,and is then connected to a block of mass $M$ as shown in the figure. If the system is released from rest,then the speed acquired by the block after it has fallen through a distance $h$ is

An annular disk of mass $M$,inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$,as shown in the figure. At some time,an impulse $J_0 \hat{x}$ is applied at a height $h$ above the center of the disk. If $h=h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement$(s)$ is(are) correct?
$(A)$ For $\mu \neq 0$ and $a \rightarrow 0, h_m=b / 2$
$(B)$ For $\mu \neq 0$ and $a \rightarrow b, h_m=b$
$(C)$ For $h=h_m$,the initial angular velocity does not depend on the inner radius $a$.
$(D)$ For $\mu=0$ and $h=0$,the wheel always slides without rolling.

$STATEMENT-1$ If there is no external torque on a body about its center of mass,then the velocity of the center of mass remains constant. because
$STATEMENT-2$ The linear momentum of an isolated system remains constant.

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes (normal to the disc and passing through the centre),and rotating with angular speeds $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. $(a)$ What is the angular speed of the two-disc system? $(b)$ Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take $\omega_{1} \neq \omega_{2}$.