Titus 2/3

Conservation of Momentum

• Definition: Conservation of momentum states that the total momentum of the universe is constant.

• Implications: Since the total momentum is constant, the change in momentum of the universe is zero.

System and Surroundings

• Defining a System: The universe can be divided into parts; the specific part we study is called the ( ext{system} ).

• Interaction with Surroundings: The system interacts with objects in its surroundings, affecting the momentum.

• Momentum Exchange: When measuring changes in momentum, the sum of changes in momentum for the system and surroundings must equal zero.

  • ( ext{Change in momentum of the system} + ext{Change in momentum of surroundings} = 0 )

  • Thus, ( ext{Change in momentum of system} = - ext{Change in momentum of surroundings} )

Vector Quantities

• Momentum is a Vector: Momentum is represented as a vector quantity, meaning it has both magnitude and direction.

• Previous Examples: Similar principles were seen in Chapter 3, Homework 1, where changes in the gravitational forces on two blocks were discussed.

Example of Two Blocks

• Analysis of Forces: Block 1 experiences a change in momentum in the positive x-direction (same as net force), while Block 2 experiences a change in momentum in the negative x-direction.

• Reciprocal Changes: The changes in momentum for both blocks are equal in magnitude and opposite in direction due to the equal and opposite forces exerted by each block on the other.

• Concept of Forces: Forces between interacting objects are equal and opposite, leading to equivalent changes in momentum.

Gravitational Force Example

• Earth and a Falling Ball: The gravitational force acting on both the ball and Earth are equal in magnitude and opposite in direction.

  • ( ext{Force on ball by Earth} = - ext{Force on Earth by ball} )

• Change in Momentum: The changes in momentum for both Earth and the falling ball are equal in magnitude, but their changes in velocity are vastly different due to their mass differences.

• Mass Ratios: Earth is significantly more massive than the ball (( ext{mass of Earth} \approx 10^{24} ) kg, while the ball could be 1 kg), leading to a negligible change in Earth's velocity compared to the ball.

Interaction Example: Ping Pong and Bowling Ball

• Collision Forces: When a ping pong ball collides with a bowling ball, the forces between them are electric forces complying with the momentum principle.

• Change in Momentum: Both balls experience equal and opposite changes in momentum due to the reciprocity of forces.

• Mass Impact: The larger mass of the bowling ball results in a minimal change in its velocity compared to the ping pong ball.

Comet in Gravitational Field

• Direction of Momentum Change: The direction of change in momentum for the comet due to gravitational force aligns with the force towards the star.

• Net Force and Change in Momentum: ( ext{Change in momentum} ) vector points toward the star, similar to the net gravitational force.

• Magnitude Consideration: The magnitude of momentum change for both the star and the comet will be the same, governed by the momentum principle and reciprocity, regardless of their mass difference.

Summary of Key Concepts

• Conservation of Momentum: The fundamental principle that the total momentum in a closed system remains constant, with exchanges occurring in an equal and opposite manner.

• Vector Nature of Momentum: Momentum is a vector quantity that must be considered in both magnitude and direction during interactions.

• Real-World Applications: Analyzing collisions and gravitational interactions helps illustrate these principles in accelerations and change in velocities.