Lecture Notes: Box Sliding Down an Inclined Plane

Introduction

• Topic: Box sliding down an inclined plane with friction.
• Goal: Find acceleration of the box and final velocity at the end of the incline.
• Given:
  • Coefficient of kinetic friction (( \mu_k )) = 0.2
  • Distance along the incline = 2 meters
  • Mass of the block = 10 kg
  • Angle of incline (( \theta )) = 30 degrees

Steps to Solve the Problem

Drawing the Free Body Diagram

• Weight: Acts downwards, component along incline - ( mg \sin \theta ), component perpendicular to incline - ( mg \cos \theta ).
• Normal Force (N): Acts perpendicular to the surface.
• Friction Force (F_k): Acts opposite to the motion.
  • Direction: Opposite to the sliding direction.

Calculating Forces

• Y-direction:
  • Forces balanced: ( N = mg \cos \theta )
• Friction Force:
  • ( F_k = \mu_k \times N
  • ( F_k = 0.2 \times 84.9 = 17 \text{ N} )

Calculating Acceleration

• X-direction: Apply Newton's Second Law:
  • ( F_x = mg \sin \theta - F_k = ma )
  • ( a = \frac{mg \sin \theta - F_k}{m} = 3.2 \text{ m/s}^2 )
  • Acceleration does not depend on mass.

Calculating Final Velocity

• Use kinematic equation:
  • ( v^2 = u^2 + 2as )
  • Initial velocity (( u )) = 0 (starts from rest)
  • ( v = \sqrt{0 + 2 \times 3.2 \times 2} = 3.58 \text{ m/s} )

Calculating Time to Reach the End

• Use equation:
  • ( v = u + at )
  • ( 3.58 = 0 + 3.2t )
  • ( t = 1.12 \text{ seconds} )

Conclusion

• Acceleration of the block is ( 3.2 \text{ m/s}^2 ).
• Final velocity at the end of the incline is ( 3.58 \text{ m/s} ).
• Time taken to reach the end is ( 1.12 \text{ seconds} ).

Note

• The terminology 'speed' and 'velocity' are used interchangeably due to linear motion.

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