Lecture Notes: Uniform Boom and Hinge Forces

Problem Overview

• Uniform Boom Weight: 700 newtons
• Object Hanging: 400 newtons
• Cable Support: Calculate parts A, B, and C related to tension and hinge forces.

Part A: Magnitude of Tension in the Cable

Method 1: Traditional X and Y Axes

• Tension Components:
  • Tension (T) points left and up in the north-west direction, creating components TY (up) and TX (left).
  • Angle: 45° (both sine and cosine of 45° can be used interchangeably in this problem).
• Weight Components:
  • Hanging weight: 400 newtons (downward).
  • Boom weight: 700 newtons (center).
• Force and Distance Components:
  • L: Length of the boom with components LX (cosine of 20°) and LY (sine of 20°).
  • For torque calculations, use perpendicular components.
• Torque Calculations:
  • Counterclockwise torque: TY x LX and TX x LY.
  • Clockwise torque: 400 x LX and 700 x L/2 x cosine of 20°.
• Solving for Tension:
  • Sums to zero: Solve for T, resulting in T ≈ 777.73 newtons.

Method 2: Tilted Axis (Using Geometry)

• Axis Alignment:
  • X-axis along the boom, Y-axis perpendicular.
  • Use angles to resolve forces into components.
• Perpendicular Forces for Torque:
  • 700 N weight: 700 x cosine of 20° for L/2.
  • 400 N weight: 400 x cosine of 20° for L.
• Tension Calculation:
  • Y-component of T (T x cosine of 25°) is perpendicular to L.
  • Solve for T, producing T ≈ 777.63 newtons (same as Method 1).

Part B: Force on the Hinge

• Net Forces:
  • Net force in X direction: TX (left) balanced by F_hinge_x (right).
  • Net force in Y direction: TY + F_hinge_y (up) balanced by 1,100 newtons (down).
• Components:
  • F_hinge_x ≈ 549.94 newtons
  • F_hinge_y ≈ 550.06 newtons
  • Use Pythagorean theorem: F_hinge = √(F_hinge_x² + F_hinge_y²) ≈ 777.82 newtons.
• Direction:
  • Angle θ: tan⁻¹(F_hinge_y / F_hinge_x) = 45°
  • Force direction: 45° above positive X-axis or northeast.

Part C: Does the Force on the Hinge Act Along the Boom?

• Conclusion:
  • No, because boom makes an angle of 20° relative to X-axis, while hinge force acts at 45°.

Key Takeaways

• Emphasize perpendicular components for torque calculations.
• Use geometry to simplify problem-solving when possible.
• Ensure understanding of force balance and vector directions.

Advice: Stay focused on identifying perpendicular forces and applying the correct trigonometric relationships for precise calculations.