Lecture Notes: Net Force and Acceleration

Overview

• Objective: Determine the magnitude and direction of net force and net acceleration on an object.
• Context: Three forces acting on an object in a specified coordinate system (x and y axes).
• Forces and angles:
  • F1: 45°
  • F2: 30°
  • F3: 35°

Steps to Solve the Problem

1. Resolving Forces into Components

• Coordinate System: Chosen with x-axis and y-axis for convenience.
• Resolve each force into x and y components.
  • F1 Components:
    • X-axis: F1 cos(45°)
    • Y-axis: F1 sin(45°)
  • F2 Components:
    • X-axis: F2 cos(30°)
    • Y-axis: F2 sin(30°) (negative direction)
  • F3 Components:
    • X-axis: F3 sin(35°)
    • Y-axis: F3 cos(35°)

2. Calculating Net Force

• X-Axis Net Force:

  • Sum: F1 cos(45°) + F2 cos(30°) - F3 sin(35°)
  • Given magnitudes: F1 = 60 N, F2 = 50 N, F3 = 40 N
  • Resultant: F_x = 62.79 N

• Y-Axis Net Force:

  • Sum: F1 sin(45°) + F3 cos(35°) - F2 sin(30°)
  • Resultant: F_y = 50.2 N

3. Resultant Force and Direction

• Magnitude of Net Force:

  • Formula: ( \sqrt{F_x^2 + F_y^2} )
  • Result: 80.4 N

• Direction (Angle θ):

  • Formula: ( \tan(θ) = \frac{F_y}{F_x} )
  • Result: θ = 38.6°

4. Calculating Net Acceleration

• Formula: Acceleration = Net Force / Mass
• Given mass = 20 kg
• Result: Acceleration = 4.02 m/s²
• Direction of Acceleration:
  • Same as net force direction, 38.6° from x-axis.

Conclusion

• Under the conditions given, the object will move along a path making a 38.6° angle with the x-axis.
• The force and acceleration are aligned in the same direction due to the vector nature of force.

Final Thoughts

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