The mass of Saturn is 5.68 x1026 kg. The radius is 5.85 x 107 m. Find the density of Saturn in kg/m3, then convert to g/cm3. Compare with the density of water, 1 g/cm3. If we could provide an ocean large enough, would Saturn sink or float?
The Correct Answer and Explanation is:
To calculate the density of Saturn, we need to use the formula for density:Density=MassVolume\text{Density} = \frac{\text{Mass}}{\text{Volume}}Density=VolumeMass
Step 1: Calculate the Volume of Saturn
Since Saturn is roughly spherical, we can calculate its volume using the formula for the volume of a sphere:V=43πr3V = \frac{4}{3} \pi r^3V=34πr3
Given:
- The radius of Saturn, r=5.85×107 mr = 5.85 \times 10^7 \, \text{m}r=5.85×107m
Plugging the radius into the formula:V=43π(5.85×107)3V = \frac{4}{3} \pi (5.85 \times 10^7)^3V=34π(5.85×107)3V≈8.50×1022 m3V \approx 8.50 \times 10^{22} \, \text{m}^3V≈8.50×1022m3
Step 2: Calculate the Density of Saturn
Now that we have the volume, we can calculate the density using the mass of Saturn m=5.68×1026 kgm = 5.68 \times 10^{26} \, \text{kg}m=5.68×1026kg:Density=5.68×1026 kg8.50×1022 m3\text{Density} = \frac{5.68 \times 10^{26} \, \text{kg}}{8.50 \times 10^{22} \, \text{m}^3}Density=8.50×1022m35.68×1026kgDensity≈6.67×103 kg/m3\text{Density} \approx 6.67 \times 10^3 \, \text{kg/m}^3Density≈6.67×103kg/m3
Step 3: Convert the Density to g/cm³
To convert the density from kg/m3\text{kg/m}^3kg/m3 to g/cm3\text{g/cm}^3g/cm3, we use the conversion factor:1 kg/m3=0.001 g/cm31 \, \text{kg/m}^3 = 0.001 \, \text{g/cm}^31kg/m3=0.001g/cm3
Thus:Density in g/cm3=6.67×103 kg/m3×0.001\text{Density in g/cm}^3 = 6.67 \times 10^3 \, \text{kg/m}^3 \times 0.001Density in g/cm3=6.67×103kg/m3×0.001Density≈6.67 g/cm3\text{Density} \approx 6.67 \, \text{g/cm}^3Density≈6.67g/cm3
Step 4: Compare with the Density of Water
The density of water is 1 g/cm³, which is much higher than the density of Saturn at 0.00667 g/cm³.
Conclusion: Will Saturn Sink or Float?
Since the density of Saturn is much less than that of water, if we could somehow provide an ocean large enough to contain the entire planet, Saturn would float. This is because objects with a density lower than the fluid they are placed in float on that fluid, similar to how ice floats on water.
