Aphotographer uses his camera, whose lens has a 50 mm focal length, to focus on an object 2.5 m away. he then wants to take a picture of an object that is 50 cm away.

part a
in which direction must the lens move to focus on this second object?

part b
how far must the lens move to focus on this second object?

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a vertical plane with constant angular velocity \omega. write down the lagrangian for a bead of mass m threaded on the rod, using r as your generalized coordinate, where r, \phi are the polar coordinates of the bead. (notice that \phi is not an independent variable since it is fixed by the rotation of the rod tobe \phi = \omega * t) solve lagrange's equation for r(t). what happens if the bead is initially at rest at the origin? if it is released from any point ro > 0, show that r(t) eventually grows exponentially. explain your results in terms of the centrifugal force m* \omega ^2 *r.