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. During the collision, why is the angular momentum conserved but not the linear momentum?

Verified step by step guidance

Understand the concept of angular momentum conservation: Angular momentum is conserved in a system if there is no external torque acting on it. In this scenario, the pivot is frictionless, meaning no external torque is applied to the gate, allowing angular momentum to be conserved.

Recognize the difference between linear and angular momentum: Linear momentum is the product of mass and velocity, while angular momentum is the product of rotational inertia and angular velocity. Linear momentum is not conserved here due to external forces acting on the system, such as the force exerted by the pivot.

Identify the system: The system consists of the wooden gate and the raven. The collision between the raven and the gate is internal to this system, and since no external torques are acting on the system, angular momentum is conserved.

Consider the forces involved: The pivot exerts a force on the gate, which affects the linear momentum of the gate. This external force means linear momentum is not conserved, as the gate's linear motion is influenced by the pivot.

Apply the principle of conservation of angular momentum: Calculate the initial angular momentum of the raven before the collision and the final angular momentum after the collision. Since no external torque is acting, the total angular momentum before and after the collision remains constant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Momentum Conservation

Angular momentum is conserved in a system when no external torques are acting on it. In the case of the gate and the raven, the pivot is frictionless, meaning no external torque is applied, allowing the angular momentum to remain constant before and after the collision. This principle is crucial for analyzing rotational dynamics in isolated systems.

Linear Momentum Conservation

Linear momentum is conserved in a system when no external forces are acting on it. However, during the collision between the raven and the gate, the pivot exerts a force on the gate, which affects the linear momentum. This external force disrupts the conservation of linear momentum, making it non-conserved in this scenario.

Collision Dynamics

Collision dynamics involve understanding how objects interact during a collision, including changes in velocity and direction. In this problem, the raven's change in velocity and direction upon bouncing back indicates an inelastic collision, affecting linear momentum but not angular momentum due to the pivot's frictionless nature. This concept helps explain the differing conservation laws in play.

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A thin uniform rod has a length of $0.500 m$ and is rotating in a circle on a frictionless table. The axis of rotation is perpendicular to the length of the rod at one end and is stationary. The rod has an angular velocity of $0.400 rad slash s$ and a moment of inertia about the axis of $3.00 \times 10^{- 3} {kg/m}^{2}$ . A bug initially standing on the rod at the axis of rotation decides to crawl out to the other end of the rod. When the bug has reached the end of the rod and sits there, its tangential speed is $0.160 m slash s$ . The bug can be treated as a point mass. What is the mass of the rod.

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Textbook 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. What is the angular speed of the gate just after it is struck by the unfortunate raven?

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Textbook Question

A small 10.0-g bug stands at one end of a thin uniform bar that is initially at rest on a smooth horizontal table. The other end of the bar pivots about a nail driven into the table and can rotate freely, without friction. The bar has mass 50.0 g and is 100 cm in length. The bug jumps off in the horizontal direction, perpendicular to the bar, with a speed of 20.0 cm/s relative to the table. What is the angular speed of the bar just after the frisky insect leaps?