Take the volume of the displaced water times it by the density of water to get the mass of the displaced water. Then times this by gravity to get the weight of the displaced water. This is also the buoyant force. Compare this number to the weight of the wooden block. How do they compare? Find the percent difference between the buoyant force and the weight of the wooden block. (They should be close though not exact). How does the calculated density of your wooden block compare to the density of water? What is the specific gravity of the wooden block? Approximately how much of the wooden block remained above the water level? Below? Times the specific gravity by 100 to get a percentage, how does your percent of block below the water level compare to the percent of specific gravity
The Correct Answer and Explanation is:
Correct Answer:
Let’s consider a wooden cube with side length 10 cm and density of wood = 0.75 g/cm³.
Let’s calculate step-by-step:
1. Volume of wooden block (and displaced water):V=10 cm×10 cm×10 cm=1000 cm3V = 10 \, \text{cm} \times 10 \, \text{cm} \times 10 \, \text{cm} = 1000 \, \text{cm}^3V=10cm×10cm×10cm=1000cm3
2. Mass of displaced water:Density of water=1.00 g/cm3⇒Mass of displaced water=Density×Volume displaced\text{Density of water} = 1.00 \, \text{g/cm}^3 \Rightarrow \text{Mass of displaced water} = \text{Density} \times \text{Volume displaced}Density of water=1.00g/cm3⇒Mass of displaced water=Density×Volume displaced
Since only part of the block is submerged, and wood floats, the volume displaced equals the volume of the submerged portion.
To find that, we first need the weight of the block:Mass of wood block=ρwood×V=0.75 g/cm3×1000 cm3=750 g⇒Weight of block=750 g×9.8 m/s2=7350 dynes=7.35 N\text{Mass of wood block} = \rho_{\text{wood}} \times V = 0.75 \, \text{g/cm}^3 \times 1000 \, \text{cm}^3 = 750 \, \text{g} \Rightarrow \text{Weight of block} = 750 \, \text{g} \times 9.8 \, \text{m/s}^2 = 7350 \, \text{dynes} = 7.35 \, \text{N}Mass of wood block=ρwood×V=0.75g/cm3×1000cm3=750g⇒Weight of block=750g×9.8m/s2=7350dynes=7.35N
Now, since the block floats, the buoyant force equals the block’s weight:Buoyant Force=Weight of displaced water=7.35 N\text{Buoyant Force} = \text{Weight of displaced water} = 7.35 \, \text{N}Buoyant Force=Weight of displaced water=7.35N
So, the mass of displaced water is:m=7.359.8=0.75 kg=750 gm = \frac{7.35}{9.8} = 0.75 \, \text{kg} = 750 \, \text{g}m=9.87.35=0.75kg=750gVolume displaced=750 g1 g/cm3=750 cm3\text{Volume displaced} = \frac{750 \, \text{g}}{1 \, \text{g/cm}^3} = 750 \, \text{cm}^3Volume displaced=1g/cm3750g=750cm3
3. Percent difference:
Actual weight = 7.35 N
Calculated buoyant force = 7.35 NPercent Difference=∣B.F.−Weight∣Average×100=∣7.35−7.35∣7.35×100=0%\text{Percent Difference} = \frac{|\text{B.F.} – \text{Weight}|}{\text{Average}} \times 100 = \frac{|7.35 – 7.35|}{7.35} \times 100 = 0\%Percent Difference=Average∣B.F.−Weight∣×100=7.35∣7.35−7.35∣×100=0%
4. Comparison to water density & Specific gravity:Specific gravity=ρwoodρwater=0.751.00=0.75\text{Specific gravity} = \frac{\rho_{\text{wood}}}{\rho_{\text{water}}} = \frac{0.75}{1.00} = 0.75Specific gravity=ρwaterρwood=1.000.75=0.75
5. Percent submerged and above water:Percent submerged=Specific gravity×100=0.75×100=75%\text{Percent submerged} = \text{Specific gravity} \times 100 = 0.75 \times 100 = 75\%Percent submerged=Specific gravity×100=0.75×100=75%Percent above water=100%−75%=25%\text{Percent above water} = 100\% – 75\% = 25\%Percent above water=100%−75%=25%
Explanation (300 words):
To understand how buoyancy works, we begin by calculating the volume of the wooden cube, which is 1000 cm³. Since the cube is floating, only a portion of it is submerged, and the volume of water displaced equals the volume submerged. The density of water is 1.00 g/cm³, and we know the density of wood is 0.75 g/cm³.
Using the formula Mass=Density×Volume\text{Mass} = \text{Density} \times \text{Volume}Mass=Density×Volume, we determine that the wooden block has a mass of 750 grams. Its weight is therefore 750×9.8=7350750 \times 9.8 = 7350750×9.8=7350 dynes, or 7.35 N.
According to Archimedes’ principle, the buoyant force on a floating object equals the weight of the liquid displaced. Since the block is floating, its weight equals the buoyant force. We calculate the volume of water needed to provide that same force: 750 g or 750 cm³. This tells us that 750 cm³ of the cube is submerged, which is 75% of the cube’s total volume.
The specific gravity of a substance is the ratio of its density to that of water. In this case, it is 0.75, matching the 75% submerged volume. This means 25% of the wooden block remains above the water. Since the percent submerged equals the specific gravity times 100, this relationship is consistent and expected.
The calculated buoyant force and the weight of the wooden block are equal, and the percent difference is 0%. This confirms that the block is correctly floating due to equal opposing forces: weight down and buoyant force up.
