Question

a metallic wire of density d floats on horizontally in water find out the maximum radius of the wire so that the wire may not sink​

Answer 1

The wire will sink if it's radius is more than $\sqrt{\frac{m}{313.58\times l} }$

Explanation:

As we know the density of water is 997 kg/$m^{2}$.

We know, Density is equal to mass upon volume.

        ⇒  $d = \frac{m}{v}$

where, d is the given density, m is the mass of the wire (which is also fixed due to constant density) and v is the volume occupied by the wire.

We know the wire is in the shape of cylinder. So, it's volume is π x $r^{2}$ x l.

where 'π' is a constant which is 3.14, 'r' is the radius of the wire and 'l' is the length of the wire which is also constant (to solve the answer in terms of radius)  .

Hence,

      ⇒ v = π x $r^{2}$ x l\

       ⇒  d = $\frac{m}{\pi \times r^{2}\times l }$

Hence, radius is

      ⇒ r = $\sqrt{\frac{m}{3.14\times 997\times l} }$

      ⇒ r = $\sqrt{\frac{m}{3130.58\times l} }$

If radius of wire is more than above equation at given length and mass (density constant due to specific amount of material)

     r ∝ $\sqrt{\frac{m}{3130.58\times l} }$

If density of wire is more than density of water, then it sinks.