 #1
Eirik
 12
 2
 Homework Statement:

Hi! This isn't really homework, but I'm practicing for my exam in mechanical physics and I'm really struggling with this one question!
Two masses, m, are rotating around the z axis as shown in the figure on a frictionless surface, but are being pulled down by a third mass, M. Find the differential equation for the movement of the system. The masses should be considered point masses and the system initially has ##R=R_0## and ##\omega=\omega_0## It should be on the following form, where ##\alpha## and ##\beta## are constants you need to find:
 Relevant Equations:

##\ddot R + \alpha g  \frac{\beta}{R^3}=0##, differential equation I need to find
##a_r = \ddot R \omega^2R##, acceleration of mass m in the circle, given in task
Here's a diagram of what the system looks like:
So far I have figured out what the initial angular velocity is, if the system is balanced (no movement):
## \sum F_m = m*\frac{v^2}{R_0}\frac{Mg}{2}=0 ##
##m \frac{v^2}{R_0}\frac{Mg}{2}=0 ## divide both sides by m
##\omega_0 = \sqrt{\frac{Mg}{2mR_0}}##
One of the hints given for the task, was that we should consider the conservation of angular momentum:
Moment of inerty, start: ##I_0=2mR_0^2##
Angular momentum, start: ##L_0=I_0\omega_0=2mR_0^2 \sqrt{\frac{Mg}{2mR_0}} ##
I am struggling to find the equation for the angular momentum after that, however. I know that the moment of inertia should still be ##I=2mR^2##, and that ##L=I\omega##. How do I find \omega?
I also have this:
## \sum F_M = Mg2S=M*\ddot R##
##S=m* (\ddot R \omega^2R)## Really not sure if this one is correct
Any help would be very greatly appreciated!
So far I have figured out what the initial angular velocity is, if the system is balanced (no movement):
## \sum F_m = m*\frac{v^2}{R_0}\frac{Mg}{2}=0 ##
##m \frac{v^2}{R_0}\frac{Mg}{2}=0 ## divide both sides by m
##\omega_0 = \sqrt{\frac{Mg}{2mR_0}}##
One of the hints given for the task, was that we should consider the conservation of angular momentum:
Moment of inerty, start: ##I_0=2mR_0^2##
Angular momentum, start: ##L_0=I_0\omega_0=2mR_0^2 \sqrt{\frac{Mg}{2mR_0}} ##
I am struggling to find the equation for the angular momentum after that, however. I know that the moment of inertia should still be ##I=2mR^2##, and that ##L=I\omega##. How do I find \omega?
I also have this:
## \sum F_M = Mg2S=M*\ddot R##
##S=m* (\ddot R \omega^2R)## Really not sure if this one is correct
Any help would be very greatly appreciated!