Lecture Notes on Vectors and Resultant Forces

Introduction to Vectors

• Vectors: Physical quantity with both magnitude and direction.
• Representation: Shown using arrows.
  • Length of arrow = Magnitude.
  • Angle of arrow = Direction (e.g., 30 degrees from the x-axis).

Forces as Vectors

• Forces are also represented as arrows.
• Resultant Force: Sum of multiple forces represented as a single vector.
  • Shows the same effect on the object as the original forces.

Parallelogram Law of Addition

• Method to find resultant force using geometric representation.
• Essential knowledge: Sine Law and Cosine Law for solving triangles.

Example Problem 1: Finding Resultant Force Magnitude and Direction

1. Given: 700 N force and 450 N force applied to an object.
2. Steps:
   • Draw lines parallel to each force to create a parallelogram.
   • Draw a diagonal from the origin to the intersection of the lines (resultant force vector).
3. Triangle Formation:
   • Two triangles formed; choose one to solve.
4. Apply Law of Cosines:
   • Identify angles using alternate interior angles (60 degrees).
   • Calculate required angles to find the resultant force.

Example Problem 2: Finding Angle for a Given Resultant Force

1. Given: Forces of 400 N and 600 N; desired resultant force of 800 N.
2. Steps:
   • Draw parallel lines for both forces.
   • Identify angles and use the law of cosines to find theta (the angle between forces).
   • Conclusion: Theta must equal 75 degrees for 800 N resultant force.

Example Problem 3: Resultant Force Directed Vertically Upwards

1. Given: 500 N and 600 N forces, resultant force directed vertically.
2. Steps:
   • Draw parallel lines for both forces.
   • Use alternate interior angles to find required angles in the triangle.
   • Apply Law of Sines to solve for resultant force magnitude and angle.

Conclusion

• Summary of types of problems involving resultant forces and addition of forces.
• Next topic: Breaking forces into x and y components.
• Encouragement: Best of luck with studies!