Conserved Quantities in Physics

Introduction to Conserved Quantities

In physics, certain quantities remain constant in isolated systems, even as other changes occur. These are called conserved quantities. Mechanical energy is conserved in the absence of friction and external forces. Two other important conserved quantities are:

• Linear momentum (focus of this chapter)

• Angular momentum (covered in later chapters)

There are additional conserved quantities in particle physics, but this chapter focuses on linear momentum.

Linear Momentum

Definition and Properties

Every moving object possesses linear momentum, which is a vector quantity pointing in the same direction as the velocity.

• Definition: The linear momentum p of an object is given by:

• Unit:

• Since velocity is a vector, momentum is also a vector.

Total Momentum of a System

The total momentum of a system is the sum of the momenta of all objects within the system:

• Example: Car 1 has a mass of 1200 kg and is going north at 8 m/s. Car 2 has a mass of 900 kg and is going east at 10 m/s. The total momentum is the vector sum of the two cars' momenta.

Momentum in Everyday Systems

• Question: What is the total momentum of all air molecules in a room?

Momentum vs. Kinetic Energy

Both momentum and kinetic energy are related to motion, but they have distinct definitions and properties:

• Momentum: (vector)

• Kinetic Energy: (scalar)

• Momentum depends on direction; kinetic energy does not.

• Both are conserved in certain types of collisions, but not always simultaneously.

Impulse and Change in Momentum

Impulse-Momentum Relationship

When a force acts on an object over a period of time, it changes the object's momentum. The change in momentum is called impulse.

• Impulse:

• Change in momentum:

• Unit of impulse:

• For constant force:

Impulse as Area Under the Curve

For one-dimensional forces, the impulse can be visualized as the area under the force vs. time graph:

• This is analogous to how work equals the area under a force vs. displacement graph.

Examples of Impulse

• Example: A force acts on an object for 5.0 ms, imparting an impulse of 10.0 N·s.

  • A. What is the maximum force, ?

  • B. What is the average force?

• Example: A force N acts over a certain time interval. Find the impulse.

Generalized Newton's Second Law

Impulse-Momentum and Newton's 2nd Law

The impulse-momentum relationship leads to a more general form of Newton's Second Law:

• This is more general than because it allows for changing mass.

Product Rule for Changing Mass

When mass changes, use the product rule:

• This accounts for systems where mass is not constant, such as rockets expelling fuel.

Examples of Changing Mass Systems

• Examples:

  • Rocket losing mass as it burns fuel

  • Sand leaking from a moving cart

  • Raindrops merging with a falling object

• Example: A small rocket is launched straight up. At a certain instant, its velocity is 120 m/s, acceleration is 18 m/s2, mass is 48 kg, and it is losing mass at 0.50 kg/s. Find the net force on the rocket.

Applications and Conceptual Questions

Situational Analysis

• Consider different scenarios where a ball undergoes changes in momentum:

  • i. Ball moving at speed v is brought to rest.

  • ii. Ball is projected from rest to speed v.

  • iii. Ball moving at speed v is brought to rest and then projected backward to its original speed.

• Largest change in momentum occurs in case iii.

Worked Example

• Example: A 200-g tennis ball moving at 10 m/s bounces off a wall at an angle. Find the change in its momentum by considering the vector components before and after the collision.

Summary Table: Comparison of Momentum and Kinetic Energy

Key Takeaways

• Linear momentum is a vector quantity conserved in isolated systems.

• Impulse is the change in momentum caused by a net force acting over time.

• Newton's Second Law can be generalized to account for changing mass.

• Momentum and kinetic energy are both important, but only momentum is always conserved in collisions and explosions.

Additional info: These notes expand on the brief points and examples in the original file, providing full academic context and explanations suitable for college-level physics students.