Question

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. (a) What is the angular speed of the gate just after it is struck by the unfortunate raven? (b) During the collision, why is the angular momentum conserved but not the linear momentum?

Short answer

(a) The angular speed of the gate is approximately 1.71 rad/s. (b) Angular momentum is conserved due to absence of external torques.

Step-by-step solution

Identify Initial Conditions

The problem involves a gate and a raven. First, we define our system. The mass of the gate (\( m_{gate} \)) is 4.5 kg, and the side length (\( L \)) is 1.5 m. The raven (\( m_{raven} \)) has a mass of 1.1 kg and an initial velocity (\( v_{i} \)) of 5.0 m/s. It rebounds at a velocity (\( v_{f} \)) of -2.0 m/s (opposite direction).

Determine Moment of Inertia

The gate is a solid square rotating about an axis passing through its edge. The moment of inertia (\( I \)) for such a gate is calculated using the formula for a square plate about its edge: \[ I = \frac{1}{3} m_{gate} L^2 \].Plug the values into the formula: \( I = \frac{1}{3} \times 4.5 \times (1.5)^2 = 3.375 \text{ kg}\cdot\text{m}^2 \).

Apply Conservation of Angular Momentum

Angular momentum is conserved during the collision because there are no external torques. Initial angular momentum (\( L_i \)) is just from the raven, since the gate is initially at rest. \[ L_i = m_{raven} \cdot v_{i} \cdot \frac{L}{2} \].After the collision, the raven moves in the opposite direction, adding negative angular momentum. \[ L_f = I \cdot \omega + m_{raven} \cdot v_{f} \cdot \frac{L}{2} \].Set \( L_i = L_f \) for solving:\[ m_{raven} \cdot v_{i} \cdot \frac{L}{2} = I \cdot \omega + m_{raven} \cdot v_{f} \cdot \frac{L}{2} \].

Solve for Angular Velocity

Substitute the given values to solve for angular velocity (\( \omega \)).\[ 1.1 \times 5.0 \times \frac{1.5}{2} = 3.375 \cdot \omega + 1.1 \times (-2.0) \times \frac{1.5}{2} \].Calculate:\[ 1.1 \times 5.0 \times 0.75 = 3.375 \cdot \omega - 1.1 \times 2.0 \times 0.75 \].\[ 4.125 = 3.375 \cdot \omega - 1.65 \].\[ 5.775 = 3.375 \cdot \omega \].\[ \omega = \frac{5.775}{3.375} \approx 1.71 \text{ rad/s} \].

Explain Conservation of Angular vs. Linear Momentum

During the collision, angular momentum is conserved because there are no external torques affecting the system. However, linear momentum is not conserved because the pivot exerts a force on the gate, providing an external force in the horizontal direction.

Key concepts

Angular Speed Calculation

The angular speed calculation is crucial to understanding how fast the gate turns after the collision with the raven. When the raven hits the door, it imparts a certain amount of angular momentum to the gate. The gate's initial angular speed is zero since it starts at rest.
To find its angular speed (\( \omega \)), we use the conservation of angular momentum, which states that the initial angular momentum of the system must equal the final angular momentum. This involves using the moment of inertia, which we will address further, and the velocities of both the raven and the gate.
By setting the angular momentum of the raven before the collision equal to the combined angular momentum of the raven and gate after the collision, we obtain the final angular speed. The angular speed is calculated using the formula:\[ \omega = \frac{5.775}{3.375} \approx 1.71 \text{ rad/s}\]This shows how much the gate rotates in one second just after impact, expressing the change in rotational motion due to the collision.

Moment of Inertia

Moment of inertia is the rotational equivalent of mass in linear motion, representing how difficult it is to change an object's rotational state about a pivot or axis. For rigid bodies like our square wooden gate, the moment of inertia depends on mass distribution relative to the axis of rotation.
For the gate in our problem, we use the formula for a square plate rotating about an edge:\[I = \frac{1}{3} m_{\text{gate}} L^2\]This formula considers the gate's mass and distance of its mass from the pivot.
Substituting the given numbers:\[I = \frac{1}{3} \times 4.5 \times (1.5)^2 = 3.375 \text{ kg} \cdot \text{m}^2\]This value means that 3.375 kg·m² encapsulates the gate's resistance to changes in its angular speed during the collision with the raven. Understanding this helps to clarify why heavier or differently distributed objects rotate slower or faster when forces are applied.

Collision Physics

Collision physics explores the interactions between objects during impacts, providing vital insights into motion changes. In this scenario, a raven hitting a wooden gate illustrates two important ideas: impulse and rebound. The raven's initial and final velocities create a difference that changes the momentum of the system.
During the collision, the raven imparts momentum to the gate, causing it to rotate. Key aspects include:

• The raven's calculated initial velocity of 5.0 m/s, which contributes a significant forward momentum as it strikes the gate.
• Upon rebounding with a velocity of -2.0 m/s, the raven exerts a backward force, influencing the final momentum of the gate and raven as a system.

By studying such collisions, one learns how energy and momentum transfer between bodies, altering their motion, which is central to predicting post-collision outcomes.

Linear Momentum Conservation

Linear momentum conservation deals with maintaining momentum in the absence of external forces. However, when discussing the gate and raven collision, linear momentum isn't conserved. This is because an external force, specifically the force exerted by the pivot, acts on the gate.
A few key points offer insight into this concept:

• In isolated systems, where no external force is applied, the total linear momentum before and after an event remains constant.
• In this exercise, the pivot creates an external horizontal force on the gate, preventing conservation of linear momentum.
• The presence of these external forces explains why linear momentum can be lost or redirected in real-world objects like our swinging gate.

This external force affects the gate, shifting linear momentum principles and emphasizing the role of external influences in many physical interactions.