Which of the following statements are correct?
$(a)$ Centre of mass of a body always coincides with the centre of gravity of the body.
$(b)$ Centre of mass of a body is the point at which the total gravitational torque on the body is zero.
$(c)$ $A$ couple on a body produces both translational and rotational motion in a body.
$(d)$ Mechanical advantage greater than $1$ means that small effort can be used to lift a large load.

$A$ hemisphere of mass $3m$ and radius $R$ is free to slide with its base on a smooth horizontal table. $A$ particle of mass $m$ is placed on the top of the hemisphere. If the particle is displaced with a negligible velocity,find the angular velocity of the particle relative to the centre of the hemisphere at an angular displacement $\theta$,when the velocity of the hemisphere is $v$.

$A$ solid sphere and a disc of the same radius are released from the same height on an inclined plane and they reach the bottom of the plane at different times. This is due to their:

$A$ 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

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.
$(a)$ Does the law of conservation of angular momentum apply to the situation? Why?
$(b)$ Find the angular speed of the two-disc system.
$(c)$ Calculate the loss in kinetic energy of the system in the process.
$(d)$ Account for this loss.

$A$ body weighs $6$ g when placed in one pan and $24$ g when placed on the other pan of a false balance. If the beam is horizontal when both the pans are empty,the true weight of the body is ....... g.