practice problem 1
- Which is more massive on the surface of the Earth?
- Which is more massive on the surface of the Moon?
- Which is heavier on the surface of the Earth?
- Which is heavier on the surface of the Moon?
- The phrase "more massive" should be read literally as "has more mass" not "fills more space".
- The phrase "heavier" should be read as "is pulled down more strongly by gravity" not "is more dense".
This is a question about how people sometimes confuse the meanings of words. The answer to all four parts of this problem are, "They are the same".
- People sometimes confuse mass and volume when using the word "massive". A kilogram of aluminum takes up more volume than a kilogram of lead because it is less dense, but this is a question about mass, not volume. All identical masses are identical no matter what they are made of. A kilogram of aluminum on Earth has the same mass as a kilogram of lead on Earth.
- Mass is an invariant quantity — it changes for nothing. All identical masses are identical no matter where they are. A kilogram of aluminum on the moon has the same mass as a kilogram of lead on the moon.
- People sometimes confuse density and weight when using words like "heavier" or "lighter". Lead may be said to be "heavier" as a material, but this is a question about weight, not density. Weight is affected by mass and gravity so two identical massed objects in the same gravitational field have the same weight. A kilogram of aluminum on Earth has the same weight as a kilogram of lead on Earth
- Weight varies with location, so objects weigh less on the moon than on the Earth. But objects with the same mass have the same weight when located in the same place. It doesn't matter where that place is. A kilogram of aluminum on the moon has the same weight as a kilogram of lead on the moon.
The future version of this question is about how the presence of an atmosphere affects the precise measurement of weight.
practice problem 2
Start with the definition of density, replace V with s3 (the volume of a cube), substitute the given quantities in SI units, and solve.
practice problem 3
|layer||depth range (km)||mean density (kg/m3)||consistency|
|crust||0 ~ 20||2700||solid|
|mantle||20 ~ 2890||4500||plastic|
|outer core||2890 ~ 5160||?||liquid|
|inner core||5160 ~ 6370||?||solid|
- the average density of the entire Earth
- the percent of the Earth's mass located in the mantle, and
- the average density of the core.
Density is the ratio of mass to volume. Use the given mass of the Earth and calculate its volume from the radius and the equation for the volume of a sphere.
ρearth = mearth Vearth ρearth = 3m 4πr3earth ρearth = 3(5.97 × 1024 kg) 4π(6.37 × 106 m)3 ρearth = 5510 kg/m3
You may want to do this is stages. Calculate the volume of the spherical shell that is the mantle.
Vmantle = 4 πr3outer − 4 πr3inner 3 3 Vmantle = 4 π ( r3outer − r3inner ) 3 Vmantle = 4 π [ (6,370,000 m)3 − (3,480,000 m)3 ] 3 Vmantle = 9.06164… × 1020 m3
Mass is density times volume. Use the density given in the data table and the mass calculated above.
mmantle = ρV mmantle = (4500 kg/m3)(9.06164… × 1020 m3) mmantle = 4.07774… × 1024 kg
Divide the mass of the mantle by the mass of the Earth to determine the percentage of Earth's mass in its most substantial layer.
mmantle = 4.07774… × 1024 kg mearth 5.97 × 1024 kg mmantle = 0.683038… ≈ 70% mearth
This part could also be solved in stages, but let's try a different approach. Start from the basic definition of density, replace mass and volume with their respective fractions of the Earth's total, then multiply by the average density. Recall that the volume of a sphere is proportional to the cube of its radius.
ρcore = mcore Vcore ρcore = (1 − 0.683038…)mearth ⎛
6370 km − 2890 km ⎞3
Vearth 6370 km ρcore = 1.94396… ρearth ρcore = (1.94396…)(5514.01… kg/m3) ρcore = 10,700 kg/m3
This number seems a bit low to me. The Earth's core is homogeneous chemically, but not physically. Both layers are a mixture of iron and nickel, but one is liquid and the other solid. The liquid outer core is reported to range in density from 10,000 to 12,000 kg/m3 and the solid inner core from 12,300 to 12,600 kg/m3. My suspicion is that the mean value is more like 10,900 kg/m3, but I don't see any error in the solution presented above. I leave you with the classic excuse found in many college math textbooks, but with a slight modification. "The disproof is left to the reader."
practice problem 4