- #1
skrat
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Homework Statement
Around vertical axis ##O## a body on picture below (see attachment) is being rotated with constant angular velocity ##\Omega ##. On the circle we have a body with mass ##m##, that can feels no friction. Find position of that body as function of ##\phi ## and time. Calculate the energy of the system. Is it constant? Also find equilibrium position and frequency for small movements around equilibrium position.
Homework Equations
The Attempt at a Solution
Here is how I started, hopefully not completely wrong.
Firstly, one inertial coordinate system with origin in axis ##0## and axis ##\hat{i} ## pointing to the right and axis ##\hat{j} ## pointed up. Second, non-inertial system positioned in center of circle with radius ##R## where one axis is parallel to the line ##l## and the other of course perpendicular the ##l##.
I used notation ##^{'}## for all coordinates in non-inertial system.
Now ##m{\ddot{r}}'=a_{rel}+2\vec{\Omega }\times v_{rel}+\vec{\Omega }\times(\vec{\Omega }\times {r}')=\vec{F_r}+\vec{F_g}##
where:
##{r}'=R(cos\phi ,sin\phi )##
##\dot{{r}'}=v_{rel}=R\dot{\phi }(-sin\phi ,cos\phi )##
##\ddot{{r}'}=R\ddot{\phi }(-sin\phi ,cos\phi )+R\dot{\phi }(-cos\phi ,-sin\phi )##
##\vec{\Omega }=(0,0,\Omega )##
Now ##2\vec{\Omega }\times v_{rel}=-2\Omega R\dot{\phi}(cos\phi ,sin\phi )## and ##\vec{\Omega }\times(\vec{\Omega }\times {r}')=-\Omega ^2R(cos\phi ,sin\phi )##.
Also ##\vec{F_r}##, which is force on body in radial direction is than ##\vec{F_r}=F_r(cos\phi ,sin\phi )##. Now ##\vec{F_g}=F_g\hat{j}=F_g(sin\Omega t \hat{{i}'}+cos\Omega t \hat{{j}'})=F_g(sin\Omega t ,cos\Omega t )##.
Now writing everything in each directions should give me:
##\hat{{i}'}##: ##-R\ddot{\phi }sin\phi -R\dot{\phi }cos\phi-2\Omega R\dot{\phi }cos\phi -\Omega ^2Rcos\phi =F_rcos\phi +F_Gsin\Omega t## and
##\hat{{j}'}##: ##R\ddot{\phi }cos\phi -R\dot{\phi }sin\phi-2\Omega R\dot{\phi }sin\phi -\Omega ^2Rsin\phi =F_rsin\phi +F_Gcos\Omega t##
That is IF I am not completely mistaken. Before I continue: So, my question here is: Is everything ok so far? Is this the right way to solve the problem OR are there easier ways? Maybe using Lagrangian mechanics or...? Thanks in advance.