$A$ large ship can float but a steel needle sinks because of

The weight of a body in water is one-third of its weight in air. The density of the body is ....... $g/cm^3$. (in $.5$)

$A$ spring balance $A$ reads $2 \,kg$ with a block of mass $m$ suspended from it. Another balance $B$ reads $3 \,kg$ when a beaker with a liquid is put on its pan. The two balances are now so arranged that the hanging mass $m$ is fully immersed inside the liquid in the beaker as shown in the figure. In this situation,

Two cylinders of same cross-section and length $L$ but made of two materials of densities $d_1$ and $d_2$ are cemented together to form a cylinder of length $2L$. The combination floats in a liquid of density $d$ with a length $L/2$ above the surface of the liquid. If $d_1 > d_2$,then:

$Assertion :$ The buoyant force on a submerged rigid object can be considered to be acting at the centre of mass of the object.
$Reason :$ For a rigid body,a force field distributed uniformly through its volume can be considered to be acting at the centre of mass of the body.

$A$ leak-proof cylinder of length $1 \; m,$ made of a metal which has a very low coefficient of expansion,is floating vertically in water at $0^{\circ} C$ such that its height above the water surface is $20 \; cm.$ When the temperature of water is increased to $4^{\circ} C,$ the height of the cylinder above the water surface becomes $21 \; cm.$ The density of water at $T=4^{\circ} C,$ relative to the density at $T=0^{\circ} C,$ is close to