AGP
Feb9-11, 12:47 PM
1. The problem statement, all variables and given/known data
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.
2. Relevant equations
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
3. 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!
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
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.
2. Relevant equations
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
3. 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!
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution