Lecture Notes: Minimum Force Required to Move a Block

Introduction

• Discussion on the minimum force required to pull a block on a rough surface with a given coefficient of friction.

Problem Setup

• A block of mass M is placed on a rough surface with coefficient of friction μ.
• The block is pulled with a force F at an angle θ from the horizontal.

Forces Acting on the Block

1. Horizontal Forces

   • Forward force component: F cos θ
   • Friction force opposing the motion.

2. Vertical Forces

   • Weight of the block: mg
   • Normal force exerted by the surface: N
   • Upward component of force: F sin θ

Equations of Motion

• In the vertical direction:

  • N + F sin θ = mg
  • Rearranging gives: N = mg - F sin θ

• In the horizontal direction:

  • F cos θ - f_s = ma
  • Where f_s is the static friction force.

Static Friction

• Maximum static friction: f_s(max) = μN
• To find minimum force F to just move the block:
  • Set static friction equal to pulling force component:
    • F cos θ = f_s(max)
    • Thus, F cos θ = μN

Calculating Minimum Force

• Substitute N from vertical equation into the friction equation:

  • F cos θ = μ(mg - F sin θ)
  • Rearranging gives:
  • F cos θ + μF sin θ = μmg

• Therefore,

  • F = (μmg) / (cos θ + μsin θ)

Angle of Pulling

• To minimize F, differentiate the function with respect to θ and set the derivative to zero:
  • Resulting condition leads to: tan θ = μ
  • Hence optimal angle to pull for minimum force is θ = tan⁻¹(μ).

Conclusion

• For minimum force required to just move the block without slipping:
  • Angle to pull: tan⁻¹(μ)
  • Minimum force: F ≈ (μmg) / sqrt(1 + μ²)

Example Problem

• Given conditions:
  • Mass of block M = 10 kg
  • Coefficient of friction μ = 0.75

1. Calculate angle: θ = tan⁻¹(0.75)
2. Minimum force required:
   • F ≈ (μmg) / sqrt(1 + μ²)
   • Calculate:
     • For M = 10 kg, g = 9.8 m/s²
     • Complete calculation for F.

---

• This problem involves applying principles of dynamics and static friction to determine the forces necessary to overcome resistance due to friction to initiate motion.
• Understanding this can aid in solving complex mechanics problems later.