Question 7.SP.35: Two small spheres, each with a mass of 2 kg, are connected b......
Two small spheres, each with a mass of 2 kg, are connected by a light string as shown in Fig. 7-32. The horizontal platform is rotating under no external moments at 36 rad/s when the string breaks. Assuming no friction between the spheres and the groove in which they ride, determine the angular speed of the system when the spheres hit the outer stops. The moment of inertia I for the disk is 0.5 kg⋅m².
The initial moment of inertia I_{i} is that of the disk and the two spheres at distance 0.075 \mathrm{~m} from the center. Hence,
I_{i}=0.5+2(2)\left(0.075^{2}\right)=0.5225 \mathrm{~kg} \cdot \mathrm{m}^{2}
The initial angular momentum is I_{i} \omega_{i} , or 0.5225(36) . The final moment of inertia I_{f} is
I_{f}=0.5+2(2)\left(0.275^{2}\right)=0.8025 \mathrm{~kg} \cdot \mathrm{m}^{2}
Since no external moments act on the system, the angular momentum is conserved; then
I_{f} \omega_{f}=I_{i} \omega_{i} \quad 0.8025 \omega_{f}=0.5225(36) \quad \therefore \omega_{f}=\underline{23.4 \mathrm{rad} / \mathrm{s}}