Is the Maximum Radius of a Wire Floating on Liquid Affected by Surface Tension?

[Amith2006]

Sir,
The material of a wire has a density of 1.4 gm/cc. If it is not wetted by a liquid of surface tension of 44 dyne/cm, then what is the maximum radius of the wire that can float on the surface of the liquid?
Can you give a hint to solve this problem?

 

[Andrew Mason]

Sir,
The material of a wire has a density of 1.4 gm/cc. If it is not wetted by a liquid of surface tension of 44 dyne/cm, then what is the maximum radius of the wire that can float on the surface of the liquid?
Can you give a hint to solve this problem?

What is the force of gravity (per cm) on the wire as a function of its radius? What is the force (per cm) required to overcome the surface tension?

AM

 

[Amith2006]

Sir,
The material of a wire has a density of 1.4 gm/cc. If it is not wetted by a liquid of surface tension of 44 dyne/cm, then what is the maximum radius of the wire that can float on the surface of the liquid?
Can you give a hint to solve this problem?

I solved it in the following way:
Let the radius, density and length of the wire be r, d and l respectively. Let T be surface tension.
Mass per unit length of the wire = volume x density
= (pi) x (r^2) x d
Force per unit length of the wire = (pi)(r^2)d x g (g ~ 1000 cm/sec^2)
By definition of surface tension,
Force per unit length of the wire = surface tension
(pi)(r^2)d x g = T
r^2 = T/[(pi) x d x g]
By solving I get,
r =1/10 cm
But the book answer is 10/7 cm. Is there any mistake in my solution?

 

[Andrew Mason]

I solved it in the following way:
Let the radius, density and length of the wire be r, d and l respectively. Let T be surface tension.
Mass per unit length of the wire = volume x density
= (pi) x (r^2) x d
Force per unit length of the wire = (pi)(r^2)d x g (g ~ 1000 cm/sec^2)
By definition of surface tension,
Force per unit length of the wire = surface tension
(pi)(r^2)d x g = T
r^2 = T/[(pi) x d x g]
By solving I get,
r =1/10 cm
But the book answer is 10/7 cm. Is there any mistake in my solution?

Your answer appears to be correct. A metal wire 1.4 cm thick will not float on water. Again, your book is wrong.

AM