Max Diameter for a Steel Rod to Float on Water: Surface Tension Calculation

[aktanuku]

Homework Statement

Consider placing a short length of small diameter steel (specific weight =490 lb/ft^3) rod on the surface of water. What is the maximum diameter that the rod can have before it will sink? Assume the surface tension forces act vertically upward.

Homework Equations

volume of cylinder = pi*r^2h

The Attempt at a Solution

I don't know if this is correct.
I am assuming the rod is lying flat on the water not standing up (like this: _ ; not like this: |) also i am assuming that the rod is half way submerged so surface tension forces are acting on on exactly half of the surface area of the rod. So the surface area of a cylinder (excluding the top and bottom) is given by 2*pi*r*h. Since i want half of that i should get pi*r*h. assume a coefficient of surface tension sigma. surface tension=sigma*pi*r*h
Since the rod has a specific weight of 490 lb/ft^3 we can multiply this by the volume of a cylinder (pi*r^2*h) and get the downward force.
Since the rod is floating these forces must be in equilibrium. Thus,

pi*r^2*h*490 = sigma*pi*r*h

so r= sigma/490

 

[KataKoniK]

This solution is partially correct. You are correct in stating that the surface tension forces act vertically upward, but the rod is not necessarily half submerged. The maximum diameter of the rod before it sinks will depend on the weight of the rod and the surface tension forces acting on it. The surface tension force can be calculated using the equation F = σL, where σ is the surface tension coefficient and L is the length of the surface in contact with the water.

To find the maximum diameter, we can set the weight of the rod equal to the surface tension force. The weight of the rod can be calculated using the density of steel (490 lb/ft^3) and the volume of the rod (πr^2h).

Therefore, we can set the equations equal to each other:

490πr^2h = σL

Solving for r, we get:

r = √(σL/490πh)

So, the maximum diameter of the rod would be the value of r that satisfies this equation. Keep in mind that this is a simplified answer and does not take into account the weight of the rod itself, which would slightly decrease the maximum diameter.