(a) 2.75 rev/min
The moment of inertia of the rod rotating about its center is:
![I_R=\frac{1}{12}ML^2](https://tex.z-dn.net/?f=I_R%3D%5Cfrac%7B1%7D%7B12%7DML%5E2)
where
is its mass
L = 0.450 m is its length
Substituting,
![I_R=\frac{1}{12}(3.30\cdot 10^{-2})(0.450)^2=5.57\cdot 10^{-4} kg m^2](https://tex.z-dn.net/?f=I_R%3D%5Cfrac%7B1%7D%7B12%7D%283.30%5Ccdot%2010%5E%7B-2%7D%29%280.450%29%5E2%3D5.57%5Ccdot%2010%5E%7B-4%7D%20kg%20m%5E2)
The moment of inertia of the two rings at the beginning is
![I_r = 2mr^2](https://tex.z-dn.net/?f=I_r%20%3D%202mr%5E2)
where
m = 0.200 kg is the mass of each ring
is their distance from the center of the rod
Substituting,
![I_r=2(0.200)(5.20\cdot 10^{-2})^2=1.08\cdot 10^{-3} kg m^2](https://tex.z-dn.net/?f=I_r%3D2%280.200%29%285.20%5Ccdot%2010%5E%7B-2%7D%29%5E2%3D1.08%5Ccdot%2010%5E%7B-3%7D%20kg%20m%5E2)
So the total moment of inertia at the beginning is
![I_1=I_R+I_r = 5.57\cdot 10^{-4}+1.08\cdot 10^{-3}=1.64\cdot 10^{-3}kg m^2](https://tex.z-dn.net/?f=I_1%3DI_R%2BI_r%20%3D%205.57%5Ccdot%2010%5E%7B-4%7D%2B1.08%5Ccdot%2010%5E%7B-3%7D%3D1.64%5Ccdot%2010%5E%7B-3%7Dkg%20m%5E2)
The initial angular velocity of the system is
![\omega_1 = 35.0 rev/min](https://tex.z-dn.net/?f=%5Comega_1%20%3D%2035.0%20rev%2Fmin)
The angular momentum must be conserved, so we can write:
(1)
where
is the moment of inertia when the rings reach the end of the rod; in this case, the distance of the ring from the center is
![r=\frac{0.450 m}{2}=0.225 m](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B0.450%20m%7D%7B2%7D%3D0.225%20m)
so the moment of inertia of the rings is
![I_r=2(0.200)(0.225)^2=0.0203 kg m^2](https://tex.z-dn.net/?f=I_r%3D2%280.200%29%280.225%29%5E2%3D0.0203%20kg%20m%5E2)
and the total moment of inertia is
![I_2 = I_R + I_r =5.57\cdot 10^{-4} + 0.0203 = 0.0209 kg m^2](https://tex.z-dn.net/?f=I_2%20%3D%20I_R%20%2B%20I_r%20%3D5.57%5Ccdot%2010%5E%7B-4%7D%20%2B%200.0203%20%3D%200.0209%20kg%20m%5E2)
Substituting into (1), we find the final angular speed:
![\omega_2 = \frac{I_1 \omega_1}{I_2}=\frac{(1.64\cdot 10^{-3})(35.0)}{0.0209}=2.75 rev/min](https://tex.z-dn.net/?f=%5Comega_2%20%3D%20%5Cfrac%7BI_1%20%5Comega_1%7D%7BI_2%7D%3D%5Cfrac%7B%281.64%5Ccdot%2010%5E%7B-3%7D%29%2835.0%29%7D%7B0.0209%7D%3D2.75%20rev%2Fmin)
(b) 103.0 rev/min
When the rings leave the rod, the total moment of inertia is just equal to the moment of inertia of the rod, so:
![I_2 = I_R = 5.57\cdot 10^{-4}kg m^2](https://tex.z-dn.net/?f=I_2%20%3D%20I_R%20%3D%205.57%5Ccdot%2010%5E%7B-4%7Dkg%20m%5E2)
So using again equation of conservation of the angular momentum:
We find the new final angular speed:
![\omega_2 = \frac{I_1 \omega_1}{I_2}=\frac{(1.64\cdot 10^{-3})(35.0)}{5.57\cdot 10^{-4}}=103.0 rev/min](https://tex.z-dn.net/?f=%5Comega_2%20%3D%20%5Cfrac%7BI_1%20%5Comega_1%7D%7BI_2%7D%3D%5Cfrac%7B%281.64%5Ccdot%2010%5E%7B-3%7D%29%2835.0%29%7D%7B5.57%5Ccdot%2010%5E%7B-4%7D%7D%3D103.0%20rev%2Fmin)