In this physics tutorial, we explore a scenario involving three blocks on a surface, where they all start from rest and accelerate together at a rate of 0.500 m/s². The blocks have the following masses: Block 1 has a mass of M, Block 2 has a mass of 2M, and Block 3 also has a mass of 2M. Our objective is to determine the coefficient of kinetic friction (μₖ) between Block 2 and the surface on which it rests.

Solution Approach:

1. Set Up and Notation:

• • We denote the tension in the string between Block 1 and Block 2 as T₁₂, and the tension between Block 2 and Block 3 as T₂₃.
  • We establish a positive direction for acceleration and forces: rightward for horizontal movements and downward for vertical movements.

1. Applying Newton’s Second Law:

• For Block 3 (M3):

-m₃g + T₂₃ = m₃(-a)

• • Here, m₃ is experiencing gravitational force downward and tension upward, with negative acceleration as it moves downward.
• For Block 2 (M2):

T₂₃ - μₖ m₂g - T₁₂ = m₂a

• • In this equation, T₂₃ is pulling to the right, kinetic friction and T₁₂ are acting to the left.
• For Block 1 (M1):

T₁₂ - m₁g = m₁a

• • T₁₂ pulls upward while gravity acts downward.

1. Solving the Equations:

• We combine the equations to eliminate tensions and solve for μₖ:
• -m₃g + μₖ m₂g + T₁₂ = -a(m₃ + m₂)
• Substituting m₃ = 2M and m₂ = 2M, we simplify and solve for μₖ:
• μₖ = (g - 5a) / 2g
• Substituting a = 0.500 m/s² and g = 9.8 m/s², we find:
  • μₖ = 0.372

Key Takeaways:

• The method demonstrates a practical application of Newton's second law in a multi-body system with kinetic friction involved.
• It's crucial to correctly direct forces and establish consistent sign conventions for solving equations systematically.
• Understanding the relationship and impact of different masses on the system's dynamics can significantly aid in solving complex problems.

This tutorial not only guides through the step-by-step problem-solving process but also helps in visualizing the forces and movements in practical scenarios, enhancing the understanding of fundamental physics concepts