I'm having a lot of trouble figuring this one out:
Study the trajectory of a free particle of mass M released from a state of rest on a
rotating, sloping, rigid plane. The angular rotation rate is Omega, and the
angle formed by the plane with the horizontal is alpha. Friction and the centrifugal force
are negligible. What is the maximum speed acquired by the particle, and what is its
maximum downhill displacement?
I attached a figure which shows what's going on, for clarification.
The relation between absolute and relative velocities is as follows:
U = u - OMEGA*y
V = v + OMEGA*x
where U and V are the absolute velocities in the absolute X and Y directions, respectively; u and v denote the velocities in the rotating x and y frames. For unforced motion, the above simplify to:
u - OMEGA*y = 0
v + OMEGA*x = 0
The Attempt at a Solution
I'm having a lot of trouble accounting for the fact that the y-axis is not orthogonal to either the x axis or the axis of rotation. I've tried accounting for that by defining a new y-axis, y' which is orthogonal, but the math is blowing up. The motion should be forced only by gravity, but I'm having no luck with the following:
-M*g*sin(alpha) = M*du/dt - fv
It works out via system of linear ordinary equations (by my math) to:
d^2v/dt^2 + f^2v +g*f*sin(alpha) = 0
Which seems far more complicated that it should at this level. Thanks in advance for any help!