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.

Practice the material and concepts discussed in Chapter 10.