"James Stokes" wrote in message ...

A particle inside a helical tube is subject to the force of gravity mg
acting vertically down and the normal force N exerted by the pipe wall. The
normal force can be decomposed into three mutually orthogonal components; a
vertical component balancing the graviational force, a radial component
providing the centripetal force, and a component providing the tangential
acceleration along the path. The question is, how do I derive these forces
in terms of the inclination angles and the other forces?

You are having your first love afair with trigonometry. :-)

Didn't you ask about this before? IIRC, it was pointed out that you
must specify whether the particle is sliding or rolling. If it's
sliding, w/o friction, then its KE at each height is mgh, where h is
the height it has fallen ... whether falling straight or twisting
through a pipe, it doesn't matter -- pure energy conservation.

Given v(h) I suppose you can also find v(t), and what's more you can
find the accelerations necessary to keep it confined to your helix;
from these you get the normal forces.

If the particle is rolling, things are more complicated: now it may
store KE both in motion of the center of mass, and in rotation. Their
sum is now mgh, again without friction -- i.e., except for the static
rolling friction necessary to stop the thing from sliding. You have
some consistency conditions, like the angular velocity times the
radius matching the linear velocity. The normal forces as before are
given by the acceleration of the center of mass ...

Hmm... the helix is actually largely a distraction ... all you have
is a mass sliding or rolling down and inclined plane, with some
additional horizontal forces added in to to confine it to a circle (as
seen from above).