3. Conservation of linear momentum

Conservation Laws

• Conservation of Energy

  • The net external work done on a closed system equals the change in energy of that system.

• Conservation of Momentum

  • Net external force on a closed system equals the time rate of change of total linear momentum.

• Conservation of Mass

  • Total mass of a closed system remains constant.

• Systems and Forces

  • System: A collection of objects.

  • Closed System: Objects that cannot leave or enter the collection.

  • Environment: All that is external to the system.

  • External Force: A force affecting objects in the system from the environment.

  • Internal Force: A force between objects within the system.

Linear Momentum

• Linear Momentum

  • Defined as: ( ext{momentum} = ext{mass} \times ext{velocity vector} ).

  • Total Linear Momentum of a System: Vector sum of all objects' linear momentums.

• Net External Force: Vector sum of all external forces acting on the system.

• Conservation of Momentum Formula

  • ( ext{net external force} = \frac{d}{dt} ext{(total linear momentum)} )

Implications of Conservation of Momentum

• Absence of External Forces

  • If net external force equals zero, total linear momentum remains constant.

  • Derivative: ( 0 = \frac{d}{dt} \sum m_i v_i ) implies ( \sum m_iv_i ) is constant.

• Single Object of Constant Mass

  • Newton's second law relates force, mass, and acceleration: ( F = m a ).

Newton's Laws of Motion

• Law I: Every body continues in its state of rest or uniform motion unless acted upon by an external force.

• Law II: The rate of change of motion is proportional to the applied force and occurs in a direction that the force acts.

• Law III: For every action, there is an equal and opposite reaction.

• Historical Context

  • Originating from Newton's "Mathematical Principles of Natural Phenomena".

  • Notable implications in rocket motion.

Center of Mass

• Defining Center of Mass

  • Equilibrium in terms of mass distribution:

    • ( m_{sys} \mathbf{v}_{sys} = \sum m_i \mathbf{v}_i )

• Geometric Definition

  • Center of mass represented as:

    • ( r_{sys} = \frac{\sum m_i \mathbf{r}i}{m{sys}} )

Collisions and Momentum

• Total Linear Momentum during Collisions

  • Momentum remains conserved: ( \sum F_{ext} = \frac{d}{dt} \sum m_iv_i ).

  • Momentum equations in three dimensions (x, y, z).

Impulse and Momentum

• Impulse

  • Defined as: ( J = \int F(t) dt )

  • Important note: It's not the fall that kills you, but the sudden stop at the bottom.