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- An electron of mass and charge moves in a uniform -directed electric field of magnitude ,
and a uniform -directed magnetic field of magnitude . The electron is situated at the
origin at with an initial -directed velocity of magnitude . Show that the
electron traces out a cycloid of the general form

Find the values of , , , and , and sketch the electron's trajectory
in the - plane when ,
, and
.

- A particle of mass and charge moves in the - plane
under the influence of a constant amplitude rotating electric field
which is such that
and
. The particle starts at rest from the origin. Determine its subsequent motion. What shape is the particle's trajectory?

- A particle of mass slides on a frictionless surface whose height is a
function of only:
*i.e.*, . The function
is specified by the parametric equations

where is the parameter. Show that the total energy of the
particle can be written

where
. Deduce that the particle undergoes periodic
motion whose frequency is amplitude independent (even when the amplitude
is large). Demonstrate that the frequency of the motion is given by
.

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Richard Fitzpatrick
2011-03-31