Exercises

- A ball bearing is dropped down an elevator shaft in the Empire State
Building ( m, latitude N). Find the ball bearing's horizontal deflection (magnitude and direction)
at the bottom of the shaft due to the Coriolis force . Neglect air resistance. (Modified from Fowles and Cassiday 2005.)
- A projectile is fired due east, at an elevation angle , from a point
on the Earth's surface whose latitude is . Demonstrate that the projectile strikes the
ground with a lateral deflection
.
Is the deflection northward or southward?
Here,
is the Earth's angular velocity, the projectile's initial speed, and
the acceleration due to gravity. Neglect air resistance. (Modified from Thornton and Marion 2004.)
- A particle is thrown vertically upward, reaches
some maximum height, and falls back to the ground. Show that the horizontal Coriolis
deflection of the particle when it returns to the ground is opposite in direction,
and four times greater in magnitude, than the Coriolis deflection when it
is dropped at rest from the same maximum height. Neglect air resistance. (From Goldstein, et al. 2002.)
- A ball of mass rolls without friction over a horizontal plane located on the surface of the Earth.
Show that in the northern hemisphere it rolls in a clockwise sense (seen from above) around
a circle of radius
- A satellite is in a circular orbit of radius about the Earth.
Let us define a set of co-moving Cartesian coordinates, centered on the satellite, such that
the -axis always points toward the center of the Earth, the -axis in the
direction of the satellite's orbital motion, and the -axis in the direction
of the satellite's orbital angular velocity,
**. Demonstrate that the equation of motion of a small mass in orbit about the satellite are**assuming that and . Neglect the gravitational attraction between the satellite and the mass. Show that the mass executes a retrograde (i.e., in the opposite sense to the satellite's orbital rotation) elliptical orbit about the satellite whose period matches that of the satellite's orbit, and whose major and minor axes are in the ratio , and are aligned along the - and -axes, respectively. - Treating the Earth as a uniform-density, liquid spheroid, and taking rotational flattening into account, show that the variation of the surface
acceleration, , with terrestrial latitude, , is
- The Moon's mean orbital period about the Earth is approximately 27.32 days,
and its orbital motion is in the same direction as the Earth's axial rotation (whose period is
24 hours). Use this data to show that, on average, high tides at a given point on the Earth
occur every 12 hours and 27 minutes.
- Demonstrate that the mean time interval between successive spring tides is days.
- Let us model the Earth as a completely rigid sphere that is covered by a shallow ocean of negligible density. Demonstrate
that the tidal elongation of the ocean layer due to the Moon is
- Estimate the tidal heating rate of the Earth due to the Moon. Is this rate
significant compared to the net heating rate from solar radiation?
- Estimate the tidal elongation of the Sun due to the Earth.
- An approximately spherical moon of uniform density, mass , radius , and effective rigidity
, is in a circular orbit of major radius about a
spherical planet of mass . The moon rotates about an axis passing through its center of mass
that is directed normal to the orbital plane. Suppose that the moon is in a synchronous state such that its
rotational angular velocity,
, matches its orbital angular velocity,
.
Let , , be a set of Cartesian coordinates, centered on the moon, such that the -axis is normal to
the orbital plane, and the -axis is directed from the center of the moon to the center of the planet. Assuming that
the moon responds elastically to the centrifugal and tidal potentials, show that the
changes in radius, parallel to the three coordinate axes, induced by the centrifugal potential are
whereas the corresponding changes induced by the tidal potential areAssuming that these changes are additive, deduce that
- An artificial satellite consists of two point objects of mass connected by a light rigid rod of length . The
satellite is placed in a circular orbit of radius (measured from the mid-point of the rod) around a planet of mass . The
rod is oriented such that
it always points toward the center of the planet. Demonstrate that the tension in the rod is