Needle of length L and density D we will float on a liquid of surface tension T if radius is equal to

To determine the radius at which a needle of length L and density D will float on a liquid with surface tension T, we need to consider the balance of forces acting on the needle. This involves understanding how surface tension works and how it interacts with the weight of the needle.

Understanding Surface Tension

Surface tension is a property of liquids that causes the surface to behave like a stretched elastic membrane. It arises from the cohesive forces between liquid molecules. When an object is placed on the surface of a liquid, the surface tension can support the object if the weight of the object is not too great.

Forces Acting on the Needle

When the needle is floating, two main forces are at play:

• Weight of the Needle (W): This is given by the formula W = V * D, where V is the volume of the needle.
• Upward Force from Surface Tension (F): This force can be calculated using the formula F = T * P, where P is the perimeter of the needle in contact with the liquid surface.

Calculating the Volume and Perimeter

The volume of the needle can be expressed as:

V = π * r² * L

where r is the radius of the needle. The perimeter P, which is the length of the needle in contact with the surface, is given by:

P = 2 * π * r

Setting Up the Equation

For the needle to float, the upward force due to surface tension must equal the weight of the needle:

T * P = W

Substituting the expressions for W and P, we get:

T * (2 * π * r) = (π * r² * L) * D

Solving for the Radius

Now, we can simplify this equation:

2 * T * π * r = π * r² * L * D

Dividing both sides by π (since it appears in both terms), we have:

2 * T * r = r² * L * D

Rearranging gives us:

r² * L * D - 2 * T * r = 0

Using the Quadratic Formula

This is a quadratic equation in terms of r. We can apply the quadratic formula:

r = [2 * T ± sqrt((2 * T)² + 4 * L * D * 0)] / (2 * L * D)

Since we are looking for a positive radius, we take the positive root:

r = (2 * T) / (L * D)

Final Expression

Thus, the radius r at which the needle will float on the liquid is given by:

r = (2 * T) / (L * D)

This formula shows that the radius of the needle is directly proportional to the surface tension of the liquid and inversely proportional to both the length of the needle and its density. In practical terms, if you increase the surface tension, the needle can be thinner and still float, while a denser needle or a longer needle would require a larger radius to achieve the same floating condition.