The Physics
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Opus in profectus

Gravity of Extended Bodies

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Problems

practice

  1. Almost all stars will die as hot, dense lumps, but some of them decide to get fat and cold for awhile first. In 5 billion years our sun will expand into a red giant (red because it will be much cooler) swallowing up Mercury, Venus, Earth, and (possibly) Mars. A few hundred million years after that, it will begin shrinking until it becomes a white dwarf (white because it will be much, much hotter) about as small as the Earth but still retaining much of it mass.

    Merging stars

    Pairs of stars in orbit around one another are common in our galaxy — more common than lone stars like our sun. The stars never have the same mass and one will always age faster than the other. The diagram above shows a binary system where one star is a red giant and the other is a white dwarf. If the two stars are close enough, the white dwarf may begin "feeding" on the red giant. The matter torn off the surface of the red giant spirals into the white dwarf forming what is known as an accretion disk. Determine the separation between the two stars shown in this diagram. Let…
    r =  separation between the stars
    a =  radius of red giant
    b =  radius of white dwarf
    ma =  mass of red giant
    mb =  mass of white dwarf
    Ignore any centrifugal effects and state your answer using the given symbols only.
  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. Dark matter
    1. The orbital speed of the planets decreases with distance from the Sun. Why does this happen? Derive a formula that shows the relationship.

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    2. The orbital speed of the stars remains roughly constant with distance from the center of the Milky Way. (This is true for other galaxies as well.) What does this tell us about the distribution of mass in galaxies? Derive a formula that shows the relationship.

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    3. Calculate the mass of the Milky Way given an orbital speed of 230 km/s and a radius of 50,000 light years. Give your answer in solar masses (m☉ = 1.99 × 1030 kg) and compare it to the number of stars in the Milky Way (~1011).
    4. Dwarf galaxies, star clusters, and gas clouds beyond the edge of the visible galaxy have nearly the same orbital speed as the stars within visible galaxy. There is evidence that rotational speeds remain roughly constant at 220 km/s out to distances of 300,000 light years or six times the radius of the Milky Way. What is so amazing about this observation and what does it imply?
  4. 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 Nebula
    Helix Nebula, STScI
    Rosette Nebula
    Rosette Nebula, source unknown
    Horsehead Nebula
    Horsehead Nebula, 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…

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    1. density
    2. gravitational field strength
    3. gravitational potential energy per unit mass

numerical

  1. 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 a, b, c, d around the Earth represent various locations on its surface. The letter o represents the center of the Earth — our origin for this problem.

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    Compute the gravitational field strength (g) of the Moon and the Sun at several locations on the Earth. Convert all answers to μN/kg (which is the same as μm/s2).

    1. Compute the gravitational field strength of the Moon…
      1. at the center of the Earth (o)
      2. at the point on the Earth that is nearest to the Moon (a)
      3. at the point on the Earth that is farthest from the Moon (b)
    2. Compute the gravitational field strength of the Sun…
      1. at the center of the Earth (o)
      2. at the point on the Earth that is nearest to the Sun (c)
      3. at the point on the Earth that is farthest from the Sun (d)
    3. Which object exerts the greater gravitational force on the Earth — the Moon or the Sun? How many times stronger is the stronger object than the weaker object? Use the values you computed for the center of the Earth to make this comparison.
    4. Which object exerts the greater tidal force on the Earth — the Moon or the Sun? How many times stronger is the stronger object than the weaker object? Use the average difference of the extreme values you computed to make this comparison.

      Δg = 12(gnear − gfar)

    5. Compile your findings in a table like the one below.
    Gravitational forces acting on the Earth * Relative to the weaker object, which is assigned the value 1.
    quantity (symbol) Moon Sun unit
    mass (m) 0.000 × 1000 0.000 × 1000 kg
    distance from Earth (r)     m
    gravitational field from object at Earth…
    …center (gcenter)     μN/kg
    …nearest to object (gnear)     μN/kg
    …farthest from object (gfar)     μN/kg
    tidal force (Δg)     μN/kg
    relative strength of…
    …gravitational field     *
    …tidal force     *
  2. Oblate spheroid
    Compare these images of Earth and Jupiter (not to scale).

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    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) 5.97217 × 1024 kg 1.89812 × 1027 kg
    period of rotation (h) 23.935 9.9250
    radius (m)
    polar 6,356,800 66,854,000
    equatorial 6,378,100 71,492,000
    oblateness: ratio of difference
    in radii to equatorial 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 internet 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 (6,378,100 m)
      b =  earth's polar radius (6,356,800 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. 9.798 m/s2 (an average value calculated over the surface of the Earth)
      3. 9.80665 m/s2 (the defined unit known by the short name of standard gravity)

algebraic

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

calculus

  1. Given the nebula in practice problem 4, 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 the 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.