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 Homework Statement:

The picture shows two rods of negligible mass and length ##d_0##. Both of them have a ball of mass ##m## in each end and a spring of natural length ##d_0## and constant ##k##.
One of the balls of each rod is fixed, and the other one is freely to move along the rod.
Initially, the rods move with zero angular velocity.
At time ##t=t_0##, the fixed balls of each rod collide inelastically.
1) Explain using words and equations what happens after the collision.
2) Determine the final angular velocity of the system
3) Determine the maximum compression of the spring
4) write the equation of motion for one of the balls after the collision
Neglect gravity.
 Relevant Equations:

##dL/dt=0##
##L=I\omega##
Since there are no external forces, the angular momentum (##L##) and linear momentum (##P##) are conserved.
Let's call the left rod ##A## and the right one ##B##.
If all the balls were fixed, I'd write
##L_0=L_f##
##L_A+L_B=(I_A+I_B)\omega_f##
From this equation I can find the final angular velocity.
The rods "would form a larger single body" which rotates with ##\omega_f## and zero translation (because the velocity of the centre of mass is zero).
The equation of any ball would be
##n) T=mr\omega^2##
Where ##T## is the tension force and ##r## the distance between the mass and the centre or mass.
But I don't know what difference the free balls and the spring make, I think it is related to the moments of inertia of the rods, but they don't have to change since the angular velocity changes for angular momentum to be conserved; if they changed, I'd have to calculate the new moments of inertia and the final angular velocity (three unknowns, one equation).
Let's call the left rod ##A## and the right one ##B##.
If all the balls were fixed, I'd write
##L_0=L_f##
##L_A+L_B=(I_A+I_B)\omega_f##
From this equation I can find the final angular velocity.
The rods "would form a larger single body" which rotates with ##\omega_f## and zero translation (because the velocity of the centre of mass is zero).
The equation of any ball would be
##n) T=mr\omega^2##
Where ##T## is the tension force and ##r## the distance between the mass and the centre or mass.
But I don't know what difference the free balls and the spring make, I think it is related to the moments of inertia of the rods, but they don't have to change since the angular velocity changes for angular momentum to be conserved; if they changed, I'd have to calculate the new moments of inertia and the final angular velocity (three unknowns, one equation).
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