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Zorluk: Çok zorProperties of Matter

A sealed, hollow cylinder made of solid metal (density =6.0 g/cm3= 6.0 \text{ g/cm}^3) is placed in a liquid solvent (density =1.5 g/cm3= 1.5 \text{ g/cm}^3). The cylinder achieves neutral buoyancy, meaning it floats completely submerged without sinking or rising. If the total outer volume of the cylinder is 400 cm3400 \text{ cm}^3, what is the volume of the hollow cavity inside the cylinder in cm3\text{cm}^3?

Cevap: 300 cm³

Cevap

300
For the cylinder to be neutrally buoyant, its total mass divided by its total volume must equal the fluid's density of 1.5 g/cm31.5 \text{ g/cm}^3. With a total volume of 400 cm3400 \text{ cm}^3, the cylinder's total mass is 600 g600 \text{ g}. Since the entire mass comes from the metal shell, dividing this 600 g600 \text{ g} by the metal's true density (6.0 g/cm36.0 \text{ g/cm}^3) reveals that the actual metal occupies 100 cm3100 \text{ cm}^3. The remaining volume (400−100400 - 100) must be the 300 cm3300 \text{ cm}^3 hollow cavity.

Adım Adım Çözüm

1
Determine the average density required for neutral buoyancy.
Average density = 1.5 g/cm31.5 \text{ g/cm}^3
An object that floats completely submerged (neutrally buoyant) must have an overall density exactly equal to the fluid it displaces.
2
Calculate the total mass of the cylinder.
Total mass = 600 g600 \text{ g}
Mass equals the overall average density multiplied by total volume (1.5×4001.5 \times 400).
3
Calculate the volume occupied by the solid metal itself.
Metal volume = 100 cm3100 \text{ cm}^3
The solid metal volume equals the total mass divided by the solid metal's density (600/6.0600 / 6.0). The hollow cavity contributes essentially zero mass.
4
Calculate the volume of the hollow cavity.
Cavity volume = 300 cm3300 \text{ cm}^3
The cavity volume is the total outer volume minus the volume of the solid metal (400−100400 - 100).

Anahtar Kavram

Density, extensive/intensive properties, and buoyancy of composite objects
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Konular

Properties of Matter

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