The Physics
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Opus in profectus

Density

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Problems

practice

  1. Your mother gives you a kilogram of aluminum and a kilogram of lead. Both objects are solid, rectangular blocks.
    1. Which is more massive on the surface of the Earth?
    2. Which is more massive on the surface of the moon?
    3. Which is heavier on the surface of the Earth?
    4. Which is heavier on the surface of the moon?
    Special notes:
    • 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".
    A variation on this practice problem appears later in the section on buoyancy.
  2. A lucite cube has a mass of 142.5 g and a width of 4.9 cm. Determine its density in kg/m3.
    Lucite cube on electronic balance Ruler next to Lucite cube
  3. It has been known for several thousand years that the Earth is spherical (by educated people, at least). Sometime in the Second Century BCE the size of the Earth was determined (r♁ = 6,370 km). By the Nineteenth Century its mass was known (m♁ = 5.97 × 1024 kg). And in the early Twentieth Century the structure of the Earth was deduced. The Earth has three main layers: crust, mantle, and core. The crust of the is the lightest and thinnest and, like the shell of an egg, contributes little to its overall mass. The mantle is a bit more dense, substantially thicker, and contains most of the Earth's mass. The core is the densest layer (but not the most massive) and is divided into a liquid outer core and a solid inner core. The relevant data for the interior of the Earth are summarized below.
    The structure of the Earth
    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
    Determine…
    1. the average density of the entire Earth
    2. the percent of the Earth's mass located in the mantle, and
    3. the average density of the core.
  4. Write something completely different.

conceptual

  1. Mayonnaise is essentially a mixture of vegetable oil and water with a bit of egg yolk added as an emulsifier (a substance that keeps the oil and water from separating). Traditional mayonnaise has a density of about 910 kg/m3 while reduced fat, low calorie, or "light" mayonnaise has a density of about 1000 kg/m3. Why is "light" (low calorie) mayonnaise "heavier" (more dense) than traditional mayonnaise?
  2. Why does "heavy cream" have a lower density than "light cream"? Explain this apparent contradiction.

numerical

  1. Find the mass of the air contained in a room that is 16.40 m long by 4.5 m wide by 3.26 m high.

statistical

  1. Determine the mean density of the Earth, moon, and sun. (That's the mean value for each object individually, not the mean of all three objects collectively.) Compile your results in a table like the one below.
    object mass radius density (kg/m3)
    the Earth      
    the moon      
    the sun      
  2. Determine the density of the following astronomical objects arranged in order of increasing mass. (Pay attention to the units.)
    object mass radius density (kg/m3)
    the sun 1.99 × 1030 kg 696,000 km  
     
    white dwarf star 0.5 to 1.4
    solar masses    		 5000 km  
     
    neutron star 1.4 to 3
    solar masses    		 10 km  
     
    stellar black hole more than 3 solar masses 2Gm/c2
    (event horizon)    		  
     
    supermassive black hole > 106 solar masses 2Gm/c2
    (event horizon)    		  
     
    the known universe 1053 kg 13.8 × 109 light years  
     
  3. It is usually said that the moon has no atmosphere, but this is not quite true. The moon's atmosphere is nowhere near as dense as the Earth's or even that of Mars (which is pretty thin) but it is more dense than the "vacuum" of interplanetary space. The table below gives the partial densities of the constituent gases in the moon's atmosphere in particles per cubic centimeter.
    The lunar atmosphere Source: NASA
    particle (atom, molecule, ion) partial density
    name symbol mass (u) (particles/cm3) (u/cm3)
    helium 42He 4 40,000
    hydrogen H2 2 35,000
    argon 40 4018Ar 40 30,000
    neon 2210Ne 22 05,000
    argon 36 3618Ar 36 02,000
    methane CH4 16 01,000
    ammonia NH3 18 01,000
    carbon dioxide CO2 44 01,000
    oxygen O+ 16 trace n/a
    aluminum Al+ 27 trace n/a
    silicon Si+ 28 trace n/a
    phosphorous P+ 31 possible n/a
    sodium Na+ 23 possible n/a
    magnesium Mg+ 24 possible n/a
    total density →
    1. Complete the table above.
    2. Determine the density of the moon's atmosphere in kg/m3.
  4. Complete the following table with values in SI units.
    object mass (kg) radius (m) density (kg/m3)
    iron shot put 7.26
    (16 lbs)
    iron atom
    (55.847 u)
    iron atom's electron cloud
    (26 melectron)
    iron nucleus
    (55.847 u)
    (~4 fm)