To tackle this problem, we need to analyze the buoyancy and stability of the two cylinders when they are combined and floating in water. The scenario describes two cylinders of equal cross-sectional area and length L, made from materials with different densities, d1 and d2. When these cylinders are cemented together, they form a new cylinder of length 2L. The key point here is that the combined cylinder floats in water, with half of its length (L) submerged and half (L) above the surface. Let's break this down step by step.
Understanding the Setup
We have two cylinders:
- Cylinder 1: Density d1
- Cylinder 2: Density d2
When combined, they create a new cylinder of length 2L. The floating condition indicates that the weight of the submerged part of the cylinder equals the weight of the water displaced by that submerged part.
Calculating the Weight of the Combined Cylinder
The volume of each cylinder can be expressed as:
- Volume of Cylinder 1 (V1) = A * L
- Volume of Cylinder 2 (V2) = A * L
Where A is the cross-sectional area. The total volume of the combined cylinder (V_total) is:
V_total = V1 + V2 = A * L + A * L = 2A * L
The weights of the cylinders can be calculated using their densities:
- Weight of Cylinder 1 (W1) = d1 * V1 = d1 * (A * L)
- Weight of Cylinder 2 (W2) = d2 * V2 = d2 * (A * L)
The total weight (W_total) of the combined cylinder is:
W_total = W1 + W2 = d1 * (A * L) + d2 * (A * L) = (d1 + d2) * (A * L)
Buoyancy and Floating Condition
For the cylinder to float, the weight of the water displaced must equal the weight of the combined cylinder. The volume of water displaced (V_displaced) when the cylinder is floating with half of its length submerged is:
V_displaced = A * (L/2) = (A * L) / 2
The weight of the displaced water (W_water) can be calculated as:
W_water = d * V_displaced = d * (A * L) / 2
Setting Up the Equation
For equilibrium, we set the weight of the combined cylinder equal to the weight of the displaced water:
(d1 + d2) * (A * L) = d * (A * L) / 2
We can simplify this equation by canceling out A and L (assuming they are not zero):
d1 + d2 = d / 2
Analyzing the Densities
From the equation derived, we can see that the sum of the densities of the two materials (d1 and d2) must equal half the density of the water (d). Since it is given that d1 is greater than d2, we can infer that:
2 * d1 > d
and
2 * d2 < d
This means that the heavier material (d1) contributes significantly to the overall density of the combined cylinder, allowing it to float despite the lighter material (d2).
Conclusion
In summary, the combined cylinder's ability to float is determined by the relationship between the densities of the materials and the density of the water. The heavier material (d1) plays a crucial role in achieving buoyancy, while the lighter material (d2) helps balance the overall structure. This balance is essential for maintaining stability in the water. Understanding these principles of buoyancy and density is fundamental in physics and engineering applications.
