Lecture Notes: Forces on a Block on an Inclined Plane with Friction

Introduction

• Objective: Calculate the forces acting on a block on an inclined plane considering friction.
• Key Forces:
  • Applied force (F)
  • Weight component parallel to the incline (mg sin theta)
  • Friction force

Problem Setup

• Incline: Friction present, unlike the previous example.
• Forces to Determine:
  • Applied force (F)
  • Reaction force (R)

Important Concepts

• Force Components:
  • Parallel force component: F cos(theta)
  • Sum of parallel weight and friction force: mg sin(theta) + friction force
• Condition: Block on the verge of moving; larger force initiates movement up the incline.

Reaction Force and Friction

• Triangle Formed by Forces:
  • Reaction force no longer perpendicular due to friction.
  • Reaction force is a vector sum of normal and friction forces.
• Angle Relation:
  • Angle between reaction force and vertical: theta + phi
  • phi needs determination to find F and R

Calculation Steps

1. Determine Angle phi:

   • Utilize right triangle formed by normal and friction forces.
   • Tangent of phi: mu * normal force / normal force = mu
   • Angle phi: arctan(mu)
   • Given mu = 0.35, phi = 19.29 degrees

2. Calculate Total Angle:

   • Total angle (theta + phi): 25 degrees + 19.29 degrees = 44.29 degrees

3. Find Reaction Force (R):

   • Use cosine of the total angle relation: cos(theta + phi) = mg / R
   • Calculation gives R ≈ 139.7 N

4. Find Applied Force (F):

   • Use tangent relation: tan(theta + phi) = F / mg
   • Calculation gives F ≈ 97.55 N*

Verification

• Verify Calculations:
  • Calculate mg sin(theta) and maximum friction force.
  • mg sin(theta) = 42.26 N
  • Maximum friction force calculation.
  • Verify F cos(theta) matches the sum of mg sin(theta) and friction force.

Conclusion

• Successfully derived F and R using calculated angles and forces.
• Technique demonstrated for finding angles and resolving forces on an inclined plane with friction.