A worker wants to turn over a uniform, 1250N, rectanglar crate ( 2.0m x 1.0m) by pulling at 53.0° on one of its vertical sides. The floor is rough enough to prevent the crate from
slipping. (a) What pull is
needed to just start the
crate to tip? (b) How hard
does the floor push upward
on the crate?

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.