Problem 51 - University Physics with Modern Physics

Question

A uniform, 4.5 -kg, square, solid wooden gate 1.5 \(\mathrm{m}\) on each side hangs vertically from a frictionless pivot at the center of its upper edge. A \(1.1-\mathrm{kg}\) raven flying horizontally at 5.0 \(\mathrm{m} / \mathrm{s}\) flies into this door at its center and bounces back at 2.0 \(\mathrm{m} / \mathrm{s}\) 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?

Step-by-Step Solution

Answer

(a) The angular speed of the gate is 1.71 rad/s. (b) Angular momentum is conserved due to no external torques, while linear momentum is not, due to external forces at the pivot.

Step 1: Identify the Conservation Principles

Since the raven collides with the gate, we need to consider the conservation of angular momentum. For the system gate + raven, angular momentum is conserved because there is no external torque acting on the system during the collision. Linear momentum is not conserved because the pivot point exerts an external force on the system, restraining it from moving linearly.

Step 2: Calculate Initial Angular Momentum

The initial angular momentum of the system is the angular momentum of the raven before the collision since the gate is initially at rest. The raven has a mass of 1.1 kg and a velocity of 5.0 m/s. The distance from the pivot to the point of impact (center of the gate) is 0.75 m (half the gate's side length). The initial angular momentum \(L_i\) is given by \(L_i = m_r \, v_i \, r\), where \(m_r\) is the mass of the raven, \(v_i\) is the initial velocity of the raven, and \(r\) is the distance from the pivot. Thus, \(L_i = 1.1 \, \text{kg} \times 5.0 \, \text{m/s} \times 0.75 \, \text{m} = 4.125 \, \text{kg} \, \text{m}^2/\text{s}\).

Step 3: Calculate Final Angular Momentum of the Raven

After the collision, the raven bounces back with a velocity of 2.0 m/s in the opposite direction. The final angular momentum of the raven \(L_r\) is given by \(L_r = m_r \, v_f \, r\), where \(v_f\) is the final velocity of the raven. Therefore, \(L_r = 1.1 \, \text{kg} \times (-2.0 \, \text{m/s}) \times 0.75 \, \text{m} = -1.65 \, \text{kg} \, \text{m}^2/\text{s}\).

Step 4: Calculate Final Angular Momentum of the Gate

By the conservation of angular momentum, the angular momentum of the gate \(L_g\) after the collision is the initial angular momentum minus the final angular momentum of the raven: \(L_g = L_i - L_r = 4.125 \, \text{kg} \, \text{m}^2/\text{s} - (-1.65 \, \text{kg} \, \text{m}^2/\text{s}) = 5.775 \, \text{kg} \, \text{m}^2/\text{s}\).

Step 5: Calculate the Moment of Inertia of the Gate

The moment of inertia \(I\) of a uniform square gate around an axis along its top edge is calculated using \(I = \frac{1}{3} m_g a^2\), where \(m_g\) is the mass of the gate and \(a\) is the side length of the gate. Thus, \(I = \frac{1}{3} \times 4.5 \, \text{kg} \times (1.5 \, \text{m})^2 = 3.375 \, \text{kg} \, \text{m}^2\).

Step 6: Calculate the Angular Speed of the Gate

The angular speed \(\omega\) of the gate is found by \(\omega = \frac{L_g}{I}\), where \(L_g\) is the final angular momentum of the gate and \(I\) is the moment of inertia. Thus, \(\omega = \frac{5.775 \, \text{kg} \, \text{m}^2/\text{s}}{3.375 \, \text{kg} \, \text{m}^2} = 1.71 \, \text{rad/s}\).

Key Concepts

Angular Speed

Angular speed is an important concept in rotational motion, describing how fast something is spinning. It is measured in radians per second (rad/s). For a rotating object like our gate, angular speed can be thought of as the rotational equivalent of linear speed.
In the exercise, after the raven strikes the gate, the gate begins to rotate. The calculation of angular speed uses the concept of angular momentum, which was initially imparted by the incoming raven.
After obtaining the final angular momentum of the gate, the angular speed is calculated using the formula: \( \omega = \frac{L}{I} \), where \( L \) is the angular momentum, and \( I \) is the moment of inertia.
This calculation helps determine how quickly the gate is rotating right after the collision.

Moment of Inertia

The moment of inertia is like a rotational version of mass. It measures how difficult it is to change the rotational speed of an object. For a simple object like our gate, the moment of inertia depends on its mass and how that mass is distributed relative to the pivot point.
The moment of inertia is crucial when calculating rotational dynamics, as it plays a role similar to mass in linear dynamics.
For the gate in the exercise, we calculate the moment of inertia using the formula: \( I = \frac{1}{3} m_g a^2 \). Here, \( m_g \) is the mass of the gate, and \( a \) is the length of one side of the gate.

• It influences how much angular velocity an object can have for a given amount of angular momentum.
• The higher the moment of inertia, the more torque needed to change the angular speed.

This concept lets us predict the rotational behavior of objects in various settings.

Conservation Principles

Conservation principles are foundational in physics, and they help solve many problems, such as our gate and raven scenario. The principle of angular momentum conservation states that if no external torque acts on a system, its angular momentum will remain constant.
In the collision described, the total angular momentum of the raven-gate system is conserved, even though the linear momentum is not. This happens because although the pivot exerts a force, it does not exert a torque, allowing us to apply the conservation principle for angular momentum.

• Angular momentum conservation explains why the gate begins to rotate when hit, as the raven's angular momentum is transferred to the gate.
• Knowing this principle helps predict the outcome of rotational collisions and interactions efficiently.

Practicing these concepts enhances understanding and problem-solving skill in physics.

Collisions in Physics

Collisions in physics involve interactions where objects exert forces on one another for a short time. These can be elastic, where total kinetic energy is conserved, or inelastic, where kinetic energy is not conserved but momentum might be. With rotational motion involved, angular momentum is key.
In our scenario, the raven colliding with the gate demonstrates an important type of collision—one involving rotation. Although linear momentum is disrupted due to the external force at the pivot, angular momentum conservation applies.

• Studying different types of collisions helps in understanding practical scenarios, like car impacts or sports interactions.
• Rotational collisions teach us about how forces and angular momentum interact in real-world situations.

This forms a critical part of mechanics, showing the diverse applications of conservation laws.