Titus 2/19

Chapter Five Overview

• Culmination of previous chapters (2, 3, and 4).

• Focus on applying the momentum principle to various systems.

  • Systems in static equilibrium (at rest).

  • Moving systems.

  • Systems moving along curved paths.

• Goal: Solve for unknown forces using the momentum principle.

Key Concepts

Static Equilibrium

• Definition: Objects at rest with no motion change.

• Focus on calculating forces acting on the system while remaining at zero net force.

Dynamic Equilibrium

• Definition: Objects moving at constant velocity.

• Net force is also zero due to constant velocity.

Problem-Solving Strategy

1. Describe the Problem: Clearly identify what is being asked.

2. Draw a Picture: Visual representation aids understanding.

3. Write Down Important Information: Note key details and given values.

4. Define the System: Identify the system to analyze and its surroundings.

5. Draw a Free Body Diagram:

   • Include all forces affecting the system.

6. Write the Momentum Principle:

   • Use differential form:[ F_{net} = \frac{dP}{dt} ]

     • Left side (net force) is sum of forces; right side (rate of change of momentum) from motion.

7. Component Form: Break forces into x, y, (and z) components.

8. Solve algebraically for unknowns.

Momentum Principle Details

• Left Side: Net force, computed as forces acting on the system.

• Right Side: Determined from motion (change in momentum over time).

• Understanding that while both sides are numerically equal, they represent different concepts.

Direction Cosines

• Useful for converting magnitude and angles of forces into components.

• For a force vector: [ F = F_{magnitude} \cdot \text{Unit Vector} ]

  • Unit vectors represented as cosine of angles to the axes:

    • ( \cos(\theta_x) ) for x-axis, ( \cos(\theta_y) ) for y-axis, ( \cos(\theta_z) ) for z-axis.

• Example Calculation:

  • In 2D, if the angle to x-axis is given, the y component can be deduced as complementary.

Example Problems

Example 1: Load Supported by Ropes

• Scenario: 90 kg load held motionless by two ropes.

  • Rope 1: Exerts force ( F_1 = (-300, 500, 0) , N ).

  • Find force of Rope 2.

Flow:

1. Describe Problem: Load is at rest; system has zero momentum.

2. Free Body Diagram: Include forces from both ropes and gravitational pull.

3. Momentum Principle:

   • X-direction: ( F_{T2} = -F_{T1} ); gives ( F_{T2} = 300 N ) to the right.

   • Y-direction: Apply to find upward force and solve.

Example 2: Parasailing Force Analysis

• Scenario: Person and parachute (mass 120 kg) moving at 15 m/s.

  • Tension in rope = 1.8 ( \times 10^4 , N ).

  • Find force on system by air.

Flow:

1. Define System: Person and parachute.

2. Free Body Diagram: Include tension, gravitational force, and air resistance.

3. Apply Momentum Principle:

   • Zero net force due to constant speed. Break forces into components to solve for the air resistance.

Example 3: Ball Held by Springs

• Scenario: Metal ball at rest, held by two springs.

• Define system and surroundings including hand, two springs and gravity force.

Flow:

1. Free Body Diagram: Indicate forces from both springs and gravitational force downward.

2. Apply Momentum Principle:

   • Forces in y-direction evaluated against gravitational pull and spring tensions.

Conclusion

• Encourage practice using problem-solving steps outlined for success.

• Explore textbook examples for reinforcement of concepts.