Q6.11 Angular momentum (S)

Calculate:

  1. the angular momentum of the earth relative to the centre of the sun (assuming that the earth’s orbit is circular, and treating the earth as a point mass);
  2. the angular momentum of an 80 kg person standing on the earth’s equator, relative to the centre of the earth;
  3. as in (2) but sited in the Appleton Tower, Edinburgh (latitude: 56).

[Mass of earth = 5.98×1024 kg; Radius of earth = 6.37×106 m; Distance of earth from the sun = 1.5×1011 m]

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Hint

Remember that angular momentum is a vector and review Key Point 5.16.
 

Solution

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Solution

  1. The earth, mass m=6×1024 kg, is in circular orbit round the sun in an orbit of radius is r=1.5×1011 m.
    The speed of the earth along the circumference of the circle is v=rω, with
    ω=2π rad/year=2×10-7 rad s-1;
    and
    v=3×104 m s-1.
    Since $\vec{v}$ is always perpendicular to $\vec{r}$,
    \[L=mrv=2.7 \times10^{40}\,\rm kg\,m^{2}s^{-1}.\]
    Graphic - No title - angmomexa
  2. The person of mass 80 kg is in circular orbit about the centre of the earth in an orbit of radius r=6.4×106 m.

    The orbital angular velocity is

    ω=2π rad/day=7.3×10-5 rad s-1

    The orbital speed is

    v=rω=467 ms-1
    Again, $\vec{v}$ is always perpendicular to $\vec{r}$, so
    L=mrv=2.4×1011 kg m2 s-1
    The direction of $\vec{L}$ is perpendicular to both $\vec{r}$ and $\vec{v}$, along the earth’s rotational axis.

  3. The person is in circular orbit, not about the centre of the earth C, but about A (see diagram).

    The radius of this orbit is
    r=rcos56=3.58×106 m
    The orbital angular velocity about A is
    ω=2π rad/day =7.3×10-5 rad s-1
    and the speed is
    v=rω=261 m s-1
    The angular momentum about C is
    \[ \vec{L}=m\vec{r}\times\vec{v} \]
    Now $\vec{r}$ and $\vec{v}$ are always perpendicular, so
    L=mvr=1.34×1011 kg m2 s-1
    Graphic - No title - angmomexb

    The direction of $\vec{L}$ is perpendicular to both $\vec{r}$ and $\vec{v}$. In this case the direction is not constant, but changes as the earth rotates (see diagram).