The Physics
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Opus in profectus

Gravity of Extended Bodies

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Problems

practice

  1. Find the separation at which a degenerate star would suck the surface matter off of a companion star.
  2. Suppose that the Earth was an infinite flat slab of thickness t with the same mean density as the Earth. Calculate t in order that this infinite flat Earth has the same acceleration due to gravity on its surface as is found on the actual spherical Earth.
    1. Solve it once the easy way, using Gauss's law for gravity.

      ∯ g · dA = −4πGm

    2. Solve it once the hard way, using Newton's law of universal gravitation for continuous distributions of matter.
       
      g(r) = − G  ⌠⌠⌠
      ⌡⌡⌡      		 r̂ dm
      r2
    3. How does this number compare to the radius of the actual spherical earth?
  3. The word nebula (plural nebulae) means cloud in latin. In astronomy, a nebula is a diffuse collection gas and dust that looks something like a cloud. Nebulae are larger than stars, but smaller than galaxies — on the order of 10-1000 solar systems in diameter. A few representative images are shown below.
    Helix NebulaHelix Nebula, Source: STScI Rosette NebulaRosette Nebula, Source: unknown Horsehead NebulaHorsehead Nebula, Source: NOAO

    A simplified model of a nebula is a spherical collection of matter whose density varies linearly from a maximum at its center to zero at its "surface". Determine the following quantities both inside and outside such a simplified nebula in terms of its radius R, the distance from the center r, the density at the center ρ0, and fundamental constants…

    1. density
    2. gravitational field strength
    3. gravitational potential energy per unit mass
  4. The data in the text file earth.txt gives the density of the Earth at varous depths below the surface. Using data analysis software (preferably something that can do numerical integration) generate a data column for the the gravitational field strength at various depths below the surface. Remember that the value of the field is 9.8 N/kg on the surface of the Earth and 0 N/kg in the center.

numerical

  1. Prolate Spheroid
    The image below shows a typical alignment of the three most popular objects in the solar system: the sun, earth, and moon (not to scale). The letters around the Earth represent various locations on the surface of the Earth.
    A =  point closest to the moon
    B =  point farthest from the moon
    C =  point closest to the sun
    D =  point farthest from the sun

    1. Complete the following table.
    2. Use the results of your calculations to explain the effects that the moon and sun have on the shape of the Earth.
    quantity moon sun
    mass (kg) 7.348 × 1022 1.989 × 1030
    distance from earth (m) 3.844 × 108 1.496 × 1011
    gravitational field (N/kg) from object
    on earth, at earth's center (g) – –
    on earth, nearest to object (gnear) – –
    on earth, farthest from object (gfar) – –
    ratio of difference in field at opposite sides of the Earth to field at center (1:x) – –
  2. Oblate Spheroid
    Compare these images of earth and Jupiter (not to scale).

    1. Complete the following table.
    2. Use the results of your calculations to explain the effect that rotation has on the shapes of earth and Jupiter.
    quantity earth jupiter
    mass (kg) 5.9742 × 1024 1.8988 × 1027
    period of rotation (h) 23.935 9.9250
    radius (m)    
    polar 6,356,750 66,854,000
    equatorial 6,378,140 71,492,000
    oblateness: ratio of difference
    in radii to polar radius (1:x)    		 – –
    rotational speed (m/s)
    at either pole – –
    on the equator – –
    centrifugal acceleration (m/s2)
    at either pole – –
    on the equator – –
    gravitational acceleration (m/s2)
    at either pole – –
    on the equator – –
    ratio of centrifugal to gravitational
    accelerations at the equator (1:x)    		 – –

investigative

  1. Determine the apparent acceleration due to gravity at your current location on earth. Here is the general procedure for doing this.
    1. Start by determining your latitude and altitude. That information is probably available somewhere on the World Wide Web or you could measure it yourself with a GPS enabled device.
    2. The earth is an oblate spheroid. Use the following fancy formula to calculate your distance (r) from the center of the Earth…
      r = h + √ (a2 cos φ)2 + (b2 sin φ)2
      (a cos φ)2 + (b sin φ)2
      where…
      h =  your altitude
      φ =  your latitude
      a =  earth's equatorial radius (6,378,135 m)
      b =  earth's polar radius (6,356,750 m)
    3. Compute the magnitude of the gravitational acceleration at your location.
    4. Compute your distance from the Earth's axis; that is, the component of r parallel to the plane of the equator.
    5. Compute the magnitude of the centrifugal acceleration at your location.
    6. Combine the gravitational and centrifugal components to determine the apparent acceleration due to gravity (g') at your location. Determine both the…
      1. magnitude and
      2. direction (relative to r, the vector that points directly toward the center of the Earth)
      (Draw yourself a little picture before you try to answer this part.)
    7. Compare your results to the values often found in textbooks and reference tables.
      1. 9.8 m/s2 (this value with its two significant digits of accuracy should agree with your results)
      2. 9.81 m/s2 (an overly accurate value that is incorrect for many places on earth)
      3. 9.80665 m/s2 (the defined unit acceleration due to gravity)

algebraic

  1. Determine the rotation curve for…
    1. a spherical galaxy with a uniform mass distribution
    2. a very thin, disk-shaped galaxy with a uniform mass distribution

calculus

  1. Given the nebula in sample problem 3, determine the…
    1. location
    2. value
    of the maximum gravitational field in terms of its radius R, the density at the center ρ0, and fundamental constants.
  2. Determine the gravitational field and gravitational potential, inside and outside the following mass distributions…
    1. a sphere of mass M and uniform density ρ.
    2. a simplified model of the Earth, whose total mass M is evenly split between the core and the mantle — ½M for each part. (Assume the crust, oceans, and atmosphere make a negligible contribution to earth's mass.)
    3. a simplified model of the Earth consisting of a core with density 2ρ and mantle with density ρ. (Assume the crust, oceans, and atmosphere make a negligible contribution to earth's mass.)
    4. a galaxy (including dark matter halo) with a density that decreases according to the function ρ = ρ0/r2, where r is the distance from the galactic center.