Notes on Inclined Planes

Key Concepts

• Inclined Plane Forces
  • Normal Force: Acts perpendicular to the surface.
  • Weight Force: Acts downward (negative y direction), represented as mg.

Triangle Representation

• Draw a triangle based on the incline:
  • Angles:
    • Angle of incline: θ
    • Complementary angle: 90° - θ
    • If θ = 30°, then:
      • Angle adjacent to θ = 60°
      • Angles in triangle: 30°, 60°, 90°

Trigonometric Functions (SOHCAHTOA)

• Use SOHCAHTOA to resolve forces:
  • Cosine:
    • cos(θ) = adjacent/hypotenuse
    • x = mg cos(θ)
    • Normal Force: N = mg cos(θ)
  • Sine:
    • sin(θ) = opposite/hypotenuse
    • y = mg sin(θ)
    • Force down incline (Fg): Fg = mg sin(θ)

Equations to Remember

1. Normal Force:
   • N = mg cos(θ)
2. Force Down the Incline:
   • Fg = mg sin(θ)

Acceleration on a Frictionless Incline

• Only Fg acting in the x direction.
• Using Newton's second law:
  • F = ma; Fg = mg sin(θ)
• Therefore, acceleration (a) down the incline:
  • a = g sin(θ)
  • Independent of mass.

Inclined Plane with Friction

• If kinetic friction is present:
  • Fk = μ_k * N
  • Net force for acceleration:
    • Net force = Fg - Fk
    • a = g sin(θ) - μ_k g cos(θ)

Block Sliding Up the Incline

• If sliding up the incline:
  • Fg acting down the incline; Fk opposes motion.
  • Net force:
    • a = -g sin(θ) - μ_k g cos(θ)

Example Problems

Problem 1: Block Sliding Down a 30° Incline

• To find acceleration:

  • Fg = mg sin(30°)
  • a = g sin(30°) = 4.9 m/s²

• Distance = 200 m; find final speed:

  • Use kinematic equation:
    • V_final² = V_initial² + 2AD
    • V_final = √(0 + 2 * 4.9 * 200) = 44.27 m/s

Problem 2: Block Sliding Up a 25° Incline

• Part A: Find acceleration:

  • Fg = mg sin(25°)
  • a = -g sin(25°) = -4.14166 m/s²

• Part B: How far up will it go?

  • Use kinematic equation:
    • V_final² = V_initial² + 2AD
    • Distance D = 23.662 m

• Part C: Time to stop:

  • Use equation:
    • V_final = V_initial + at
    • t = 3.38 seconds

Additional Resources

• For more kinematic formulas, refer to online videos on the topic.