liquid displaced: FB = PLVg, where FB is the buoyant force, PL is the density of the liquid (not the density of the submerged object!), and V is the volume of the liquid displaced. Since (PLV) is the mass of the displaced liquid, then (PLVg) is the weight of the displaced liquid. A wooden cube with a side of 10 cm is floating on the surface of the water. The density of wood is 0.75 g/cm3 and the density of water is 1 g/cm3. Is the cube floating completely submerged in water or only partially? If only partially, what percentage of the volume is submerged? (Draw a free body diagram of the block) If a cube of unknown material sinks completely and falls to the bottom of the water, draw a free body diagram. Can we find out the density of the block from only knowing that it sinks?
Archimedes’ Principle: The buoyant force on a submerged object is equal to the weight of the volume of the liquid displaced: FB = PLVg, where FB is the buoyant force, PL is the density of the liquid (not the density of the submerged object!), and V is the volume of the liquid displaced. Since (PLV) is the mass of the displaced liquid, then (PLVg) is the weight of the displaced liquid. A wooden cube with a side of 10 cm is floating on the surface of the water. The density of wood is 0.75 g/cm3 and the density of water is 1 g/cm3. Is the cube floating completely submerged in water or only partially? If only partially, what percentage of the volume is submerged? (Draw a free body diagram of the block) If a cube of unknown material sinks completely and falls to the bottom of the water, draw a free body diagram. Can we find out the density of the block from only knowing that it sinks?
The Correct Answer and Explanation is:
Answer:
1. Wooden Cube:
The cube is only partially submerged in water.
To find the percentage of volume submerged, use Archimedes’ Principle:Fraction Submerged=Density of ObjectDensity of Liquid=0.751=0.75\text{Fraction Submerged} = \frac{\text{Density of Object}}{\text{Density of Liquid}} = \frac{0.75}{1} = 0.75Fraction Submerged=Density of LiquidDensity of Object=10.75=0.75
So, 75% of the cube’s volume is submerged.
2. Free Body Diagram (Floating Cube):
- Upward Force (Buoyant Force, FB): Acts upward and equals the weight of the displaced water.
- Downward Force (Weight of the Cube, W): Acts downward due to the mass of the cube.
Since the cube is floating at equilibrium:FB=WFB = WFB=W
This means the buoyant force balances the weight of the cube. Only part of the cube’s volume displaces water to create that force.
3. Cube of Unknown Material That Sinks:
- Free Body Diagram:
- Upward Force (Buoyant Force, FB): Due to the displaced water, but less than the weight of the object.
- Downward Force (Weight, W): Greater than FB, causing the cube to sink.
4. Can We Determine the Density of the Sunken Cube?
No, we cannot determine the exact density of the cube just by knowing that it sinks.
We only know:Density of Object>Density of Water(1 g/cm3)\text{Density of Object} > \text{Density of Water} (1 \, \text{g/cm}^3)Density of Object>Density of Water(1g/cm3)
To find the exact density, more information is needed, such as the weight of the object or the volume of water displaced before it touches the bottom (if it reaches equilibrium while sinking slowly).
Summary:
- The wooden cube floats partially, with 75% submerged.
- The sinking cube has a density greater than 1 g/cm³, but the exact density cannot be determined without additional data.
- These conclusions follow from Archimedes’ Principle, which relates buoyant force to the weight of the displaced fluid.
