Total Angular Momentum of 2 connected falling bodies.by xtinch Tags: angular momentum, kinematics, mechanics, rigid body 

#1
Jan912, 12:16 PM

P: 4

I have two 2 rigid bodies with masses m1 and m2 and Moments of Inertia I1 and I2, they are connected by a free rotational joint at some point, their coms lie at c1 and c2. There's gravity.
In the beginning both have some angular velocity [itex]\omega_i[/itex] Questions:  Total angular momentum of the system Attempt:  Calculating L of the two bodies is no Problem with [itex]L=I\cdot\omega[/itex], Also if they're unconnected L is simply [itex]L_1+L_2[/itex]. However my attempts at finding the total angular Momentum don't seem to work out.  Also I thought to calculate the total Angular Momentum I would have to shift the two Momenta to the COM via the parallel axis thm, and then build the sum. So I'm stuck  any help would be very much appreciated. Best, xtin 



#2
Jan912, 12:32 PM

P: 1,135

Hi xtinch!!
Welcome to PF !! Use [itex]I = I_{COM} + md^2[/itex] Here d is the distance b/w COM of body and axis along which you need to find I 



#3
Jan912, 12:48 PM

P: 4

I see, [itex]I=I_{COM}+md^2[/itex] is the parallel axis thm i mentioned, also I use [itex]R_{ig}*I*R_{ig}^T[/itex] to rotate the MoI.
So my final attempt was to rotate the MoI so they coincide with the global coordinate frame, then translate them to the global COM. (Let [itex]T(v)[/itex] be the Matrix that shifts the MoI by the vector v) [itex]L_{total} = R^T_1*I_1*R_1+T(c_{total}c_1)+R^T_2*I_2*R_2+T(c_{total}c_2)[/itex] I also tried [itex] R^T_1*(I_1+T(c_{total}c_1))*R_1[/itex] to no avail. =\ 



#4
Jan912, 12:51 PM

P: 1,135

Total Angular Momentum of 2 connected falling bodies.I can't really imagine the case !! 



#5
Jan912, 12:59 PM

P: 4

Here's an image, m3 is simply a mass that puts the system into rotation. I'm only interested in the Momentum after the collision. Below is a plot of the L calculated with my above suggestion  it's clearly not constant. So m1+m2 fall together as it's basically a box with an attached, freely rotateable rod (there are no collisions between the rod and the box), hit m3 which is fixed in space, start to rotate and then AngVel should be constant. (Note: ignore that the plot is labeled AngVel it's the momentum ;) ) 



#6
Jan1012, 02:02 AM

P: 4

Ok, I found the solution. I forgot to account for the rotation of the masses around the com.
The solution is simply: [itex]L_{total} = I_1\cdot\omega_1+m_1*(c_1c_{total})\times(v_{com}v_1)+I_2\cdot\omega_2+m_2*(c_2c_{total})\times(v_{com}v_2)[/itex] Thanks for your help and I hope this will help others who search for this simple but annoying thing in vain ;) 


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