The 10-g bullet having a velocity of 800 m / s is fired into the edge of the 5-kg disk as shown

step 1 In this question the mass moment of inertia IGD equals of 5 .5 .5 .4 square equals 0 .4 kilogram

The 10-g bullet having a velocity of 800 m / s is fired...

The $10-\mathrm{g}$ bullet having a velocity of $800 \mathrm{~m} / \mathrm{s}$ is fired into the edge of the $5-\mathrm{kg}$ disk as shown. Determine the angular velocity of the disk just after the bullet becomes embedded into its edge. Also, calculate the angle $\theta$ the disk will swing when it stops. The disk is originally at rest. Neglect the mass of the rod $A B$.

Key Concepts

Conservation of Angular Momentum

This principle states that if no external torques act on a system, the total angular momentum remains constant. In the context of a bullet embedding into a rotating disk, the angular momentum before the collision (resulting from the bullet's motion) equals the angular momentum just after the collision. This conservation is key to determining the post-collision angular velocity of the composite system.

Moment of Inertia

The moment of inertia quantifies how the mass of an object is distributed relative to the axis of rotation, and thus its resistance to angular acceleration. Calculating the moment of inertia for the disk and the bullet (treated as a point mass) is necessary to analyze the rotational dynamics after the collision, as it directly affects the angular velocity.

Inelastic Collisions in Rotational Systems

In an inelastic collision, such as when the bullet becomes embedded in the disk, kinetic energy is not conserved due to energy losses (e.g., as heat or deformation), but angular momentum is conserved. Recognizing this distinction is crucial when applying conservation principles to determine the immediate rotational behavior of the system after impact.

Energy Conservation in Rotational Motion

After the collision, the rotating system may convert its kinetic energy into gravitational potential energy as it swings upward. The conservation of energy principle in rotational motion allows the determination of characteristics like the maximum swing angle by equating the initial rotational kinetic energy to the gravitational potential energy at the peak of the motion.

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