A metallic wire of density d floats horizontally in water . The maximum radius of wire so that it may not sink in water will be ??

To find the maximum radius of a metallic wire so that it does not sink in water, we need to consider the buoyant force acting on the wire and the weight of the wire.

The buoyant force exerted on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. If this buoyant force is greater than or equal to the weight of the wire, the wire will float.

Let's call the maximum radius of the wire 'r'. We can assume the wire to be in the shape of a cylinder. The volume of a cylinder is given by the formula:

V = πr²h

where 'V' is the volume, 'r' is the radius, and 'h' is the height of the cylinder (which we can assume to be negligible since the wire is floating horizontally).

The weight of the wire can be calculated using the formula:

Weight = Volume × density × acceleration due to gravity

The weight of the wire is acting downward, and the buoyant force is acting upward. For the wire to float:

Buoyant force ≥ Weight of the wire

Now let's calculate the weight of the wire and find the maximum radius:

1. Calculate the weight of the wire:
Weight = V × density × g
= (πr²h) × density × g
= (πr² × 0) × density × g (Since height, 'h', is negligible)
= 0

The calculated weight of the wire is zero because the wire is floating.

2. Calculate the buoyant force:
Buoyant force = Weight of the fluid displaced
= Weight of the water displaced by the wire

To calculate the weight of the water displaced, we need to know the volume of water displaced by the wire. Since the wire is floating, the weight of the water displaced is equal to the weight of the wire (as per Archimedes' principle).

Therefore, the maximum volume of water displaced by the wire is equal to its weight.

3. Set the weight of the water displaced equal to the weight of the wire:
V × density of water × g = V × density of wire × g

Canceling out the 'g' and rearranging the equation, we get:
density of water = density of wire

This means, for the wire to float, the density of the wire should be equal to the density of water.

Finally, we can find the maximum radius of the wire by substituting the given density, 'd', for the density of the wire and the density of water.

Therefore, the maximum radius of the wire so that it does not sink in water is d.

Are you intending to ignore surface tension?

I assume not.

surface tension=force/length

force=mg=g*PI*r^2*d*length
let mu be the value of surface tension for water, and as it varies with temperature, look that up.

mu=PIr^2*length*g/length

solve for r.