A uniform, 4.5-kg, square, solid wooden gate 1.5 m on each side hangs vertically from a frictionless pivot at the center of its upper edge. A 1.1-kg raven flying horizontally at 5.0 m/s flies into this door at its center and bounces back at 2.0 m/s in the opposite direction. What is the angular speed of the gate just after it is struck by the unfortunate raven?

Verified step by step guidance

1. First, identify the conservation of angular momentum principle. The angular momentum before the collision must equal the angular momentum after the collision, as no external torques are acting on the system.
2. Calculate the initial angular momentum of the raven. Use the formula for linear momentum, \( p = mv \), where \( m \) is the mass of the raven and \( v \) is its velocity. The raven's initial angular momentum relative to the pivot is \( L_{initial} = m_{raven} \cdot v_{raven} \cdot r \), where \( r \) is the distance from the pivot to the point of impact (0.75 m, half the side of the gate).
3. Calculate the final angular momentum of the raven after it bounces back. Use the same formula, but with the final velocity of the raven. The direction of the velocity is opposite, so the angular momentum will be negative: \( L_{final} = m_{raven} \cdot v_{final} \cdot r \).
4. Determine the change in angular momentum of the raven, \( \Delta L = L_{initial} - L_{final} \). This change in angular momentum is transferred to the gate.
5. Use the formula for the moment of inertia of a square gate about the pivot point, \( I = \frac{1}{3} m_{gate} \cdot L^2 \), where \( m_{gate} \) is the mass of the gate and \( L \) is the length of the side of the gate. Then, use the relationship \( \Delta L = I \cdot \omega \) to solve for the angular speed \( \omega \) of the gate: \( \omega = \frac{\Delta L}{I} \).

Key Concepts

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torque acts on a system, the total angular momentum remains constant. In this scenario, the raven's impact on the gate is an internal interaction, so the angular momentum before and after the collision must be equal, allowing us to calculate the gate's angular speed post-collision.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. For a square gate pivoted at its upper edge, the moment of inertia can be calculated using the formula for a solid square plate. This value is crucial for determining the gate's angular speed after the collision, as it relates angular momentum to angular velocity.

Elastic Collision

An elastic collision is one where kinetic energy is conserved. In this problem, the raven bounces back after hitting the gate, indicating an elastic collision. Understanding this concept helps in analyzing the energy transfer during the collision, which affects the gate's motion and allows us to apply conservation laws effectively.