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1.
Perform
a numerical calculation using the parameters used in Example 10.2 and Figure
10.4e of the textbook, but find the initial velocity for which the path of
motion passes back over the initial position in the rotating system. At what time does the puck exit the
merry-go-round?

2.
Determine
how much greater the gravitational field strength *g* is at the pole than at the equator. Assume a spherical Earth. If the actual measured difference is Dg
= 52 mm/s_{2}, explain the difference. How might you calculate this difference
between the measured result and your calculation?

3.
If a
projectile is fired due east from a point on the surface of the Earth at a
northern latitude *l* with a velocity of magnitude *V*_{0} and at an angle of inclination to the horizontal *a*, show that the lateral deflection when the projectile
strikes Earth is

where *w* is the rotation frequency of the Earth.

4.
Consider
a particle moving in a potential *U*(*r*). Rewrite the Lagrangian in terms of a coordinate system in uniform
rotation with respect to an inertial frame. Calculate the Hamiltonian and determine
whether *H* = *E*. Is *H* a constant of motion? If *E* is not a constant of motion, why isn't it? The expression for the Hamiltonian thus obtained is the standard formula
(1/2)*mv*^{2} + *U* + an additional term. Show that the extra term is the
centrifugal potential energy.

This set covers the material discussed in Chapter 10.