Titus 2/5

Momentum Principle

• The momentum principle states that the change in momentum of an object is equal to the net force acting on it multiplied by the change in time (Δt).

• Example: For star one in a binary star system, the change in momentum is:

  • Δp (star one) = F (net on star one) × Δt

Defining the System

• Single Star System (Star One):

  • When considering just star one, it experiences a change in momentum due to a net force from its surroundings.

• Binary Star System (Both Stars):

  • When defining the system to include both stars, the net force on the entire system becomes zero (assuming no significant interactions with surroundings).

  • This implies that the change in momentum for the entire system is also zero, despite individual changes in momentum for each star.

  • The changes in momentum of the two stars cancel each other out, leading to:

    • Δp (system) = Δp (star one) + Δp (star two) = 0

Conservation of Momentum

• Key Concept:

  • Momentum change in a system results from an external net force, and the opposite occurs in the surroundings.

  • If the surroundings exert a force causing a momentum change in the system, the surroundings will experience an equal and opposite change in momentum.

  • The total momentum of the system plus surroundings equals zero, demonstrating conservation of momentum in an isolated system.

Closed System Definition

• A closed system is one where the net force acting on it is zero. In this scenario:

  • Momentum does not change due to a lack of external forces.

  • The total momentum remains constant, i.e.,

    • P(final) = P(initial)

  • Example with two stars in the system leads to:

    • Δp (star one) + Δp (star two) = 0

Example: Bowling Ball and Ping Pong Ball Collision

• Scenario:

  • A bowling ball (initially at rest) floats in space while a ping pong ball moves towards it.

  • Upon collision, the momentum change in the ping pong ball and bowling ball is analyzed as:

    • Net momentum for the system equals zero since it's a closed system.

  • The loved-up objects demonstrate separate momentum changes that balance out to zero.

Example: Colliding Lumps of Clay

• Interaction between two lumps of clay that collide and stick together:

  • System: Both lumps of clay

  • Initial conditions include mass and velocity.

  • Due to their interaction:

    • Momentum conservation states:

    • m1v1(initial) + m2v2(initial) = (m1+m2)*v(final)

  • Solving gives the final velocity after collision while ignoring external forces during the quick interaction.

Center of Mass Concept

• The center of mass represents the average position of the mass distribution in a multi-particle system.

• The momentum can be shown as:

  • p (system) = M (total mass) × v (center of mass)

• If momentum is constant, then the center of mass velocity must also be constant.

Free Body Diagrams and Net Forces

• To analyze systems:

  • Draw free body diagrams to visualize forces acting on an object (e.g., spring force vs. gravitational force).

  • Example: A mass hanging from a spring will experience upward force by the spring (F_spring) and a downward gravitational force (F_gravity), where:

    • F_net = F_spring + F_gravity = 0 (in equilibrium state)

  • This leads to:

    • k * s = mg, where s is the distance stretched and k is spring constant.

Summary of Forces in a Spring System

• In analyzing spring forces:

  • The spring behaves like a bond that can compress (repulsive) or stretch (attractive).

  • Importance of understanding these behaviors helps explain atomic and molecular bonds in solids.