Example 10.3 derived the equations of motion for a spherical pendulum. When dealing with a real pendulum, however, we are actually operating in a rotating frame. Repeat the derivation of the equations of motion, but this time, place the pendulum in a local vertical reference frame at latitude β (see Tutorial 10.1). You may use whatever reference frames and coordinate systems you consider to be appropriate, but remember that your equations of motion must reflect the motion of the pendulum in all three dimensions relative to the local vertical.
a. Using your new equations of motion, explain what effect the rotation of the earth has on the plane of the pendulum's swing.
b. French physicist Jean-Bernard-Leon Foucault (1819-1868) predicted ´ that a pendulum could be used to prove the rotation of the earth. In 1851, a 67 m pendulum with a 28 kg bob was constructed in Paris (latitude 48? N). Estimate the rate of rotation of the plane of this pendulum's swing
c. What happens if a Foucault pendulum is constructed at the north (or south) pole? At the equator?