A horizontal disk with a moment of inertia I1 is rotating...

A horizontal disk with a moment of inertia $I_{1}$ is rotating freely at an angular speed of $\omega_{1}$ when a second, nonrotating disk with a moment of inertia $I_{2}$ is dropped on it (Fig. $10-11$ ). The two then rotate as a unit. Find the final angular speed. Ignore the central rod. From the law of conservation of angular momentum, $$\begin{array}{l} \text { Angular momentum before }=\text { Angular momentum after }\\ I_{1} \omega_{1}+I_{2}(0)=l_{1} \omega+I_{2} \omega\\ \text { Solving this equation leads to } \quad \omega=\frac{l_{1} 4_{1}}{l_{1}+I_{2}} \end{array}.$$

A horizontal disk with a moment of inertia $I_{1}$ is rotating freely at an angular speed of $\omega_{1}$ when a second, nonrotating disk with a moment of inertia $I_{2}$ is dropped on it (Fig. $10-11$ ). The two then rotate as a unit. Find the final angular speed. Ignore the central rod.
From the law of conservation of angular momentum,
$$\begin{array}{l}
\text { Angular momentum before }=\text { Angular momentum after }\\
I_{1} \omega_{1}+I_{2}(0)=l_{1} \omega+I_{2} \omega\\
\text { Solving this equation leads to } \quad \omega=\frac{l_{1} 4_{1}}{l_{1}+I_{2}}
\end{array}.$$

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Key Concepts

Angular Momentum Conservation

Angular momentum is conserved in a system where no external torques act. This principle allows us to equate the initial angular momentum of individual rotating bodies with the final angular momentum of the combined system, even if other quantities like kinetic energy change due to interactions.

Moment of Inertia

The moment of inertia measures an object's resistance to changes in its rotational motion. It depends on both the mass and the distribution of that mass relative to the rotation axis. In problems involving rotation, the moments of inertia of individual components are essential in determining the dynamics of the system.

Combined System Dynamics in Rotational Motion

When two rotating bodies come together and rotate as a single unit, the total moment of inertia becomes the sum of the individual moments of inertia. The final angular speed can then be determined by applying the conservation of angular momentum to the entire system, emphasizing the interplay between the distribution of mass and rotational speed.

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