A 0.560 kg wood block is firmly attached to a very light horizontal spring – Free 88A

A 0.560 kg wood block is firmly attached to a very light horizontal spring (k = 190 N/m ) as shown in the figure(Figure 1). This block-spring system, when compressed 5.1 cm and released, stretches out 2.3 cm beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table?

Answer

How to Calculate the Coefficient of Kinetic Friction in a Spring-Block System

In this scenario, we are given a wood block attached to a horizontal spring, with the following data:

• Mass of block (m) = 0.560 kg
• Spring constant (k) = 190 N/m
• Initial compression (x₁) = 5.1 cm = 0.051 m
• Extension beyond equilibrium (x₂) = 2.3 cm = 0.023 m

We aim to find the coefficient of kinetic friction (μ_k) acting between the block and the table.

🔍 Understanding the Physical Process

When the spring is compressed and released, it propels the block forward. The block passes the equilibrium point and comes to rest after stretching 2.3 cm beyond. Since the block doesn’t return to the original point, energy must have been lost due to frictional force.

🔬 Step-by-Step Energy Analysis

The system’s mechanical energy is conserved if there’s no friction. However, here, friction does work and converts some spring potential energy into heat. Let’s apply the work-energy principle:

Work done by friction = Initial spring potential energy − Final spring potential energy

Spring potential energy (U) is given by:

U = (1/2)·k·x²

Initial energy when compressed (x₁ = 0.051 m):

U₁ = (1/2)(190)(0.051)² ≈ 0.247 J

Final energy at max extension (x₂ = 0.023 m):

U₂ = (1/2)(190)(0.023)² ≈ 0.050 J

Energy lost due to friction:

ΔE = U₁ – U₂ ≈ 0.247 – 0.050 = 0.197 J

🧮 Calculating the Work Done by Friction

Work done by kinetic friction:

W_friction = – f_k · d = – μ_k·m·g·d

Where:

• μ_k = coefficient of kinetic friction (to find)
• m = 0.560 kg
• g = 9.8 m/s²
• d = total distance traveled = x₁ + x₂ = 0.051 + 0.023 = 0.074 m

Now plug values into the energy loss equation:

0.197 = μ_k · 0.560 · 9.8 · 0.074

Calculate denominator:

0.560 × 9.8 × 0.074 ≈ 0.406 J

Solve for μ_k:

μ_k = 0.197 / 0.406 ≈ 0.485

✅ Final Answer:

The coefficient of kinetic friction (μ_k) is approximately 0.485.

🔗 Key Physics Concepts Involved

• Conservation of mechanical energy
• Spring potential energy: U = (1/2)kx²
• Work-energy theorem
• Kinetic friction: f_k = μ_kmg

📘 Applications

Understanding friction in spring-mass systems is vital in fields like engineering, material design, and mechanics. It helps improve performance in suspension systems, shock absorbers, and automation tools.