To determine the maximum distance the spring will compress when a block is dropped onto it, we can apply the principles of energy conservation. In this scenario, the gravitational potential energy of the block will be converted into the elastic potential energy stored in the spring as it compresses. Let's break this down step-by-step.

Step 1: Calculate the Initial Gravitational Potential Energy

First, we need to find the gravitational potential energy (PE) of the block just before it hits the spring. The formula for gravitational potential energy is:

PE = mgh

• m = mass of the block = 2.14 kg
• g = acceleration due to gravity = 9.81 m/s²
• h = height from which the block is dropped = 43.6 cm = 0.436 m

Now, substituting the values into the formula:

PE = 2.14 kg × 9.81 m/s² × 0.436 m

PE ≈ 9.17 joules

Step 2: Calculate the Spring Constant in Consistent Units

The spring constant (k) is given as 18.6 N/cm, but we need it in N/m for consistency. Since 1 N/cm = 100 N/m, we convert:

k = 18.6 N/cm × 100 = 1860 N/m

Step 3: Set Up the Equation for Elastic Potential Energy

The elastic potential energy stored in the spring when compressed is given by the formula:

Elastic PE = (1/2)kx²

At maximum compression, all the gravitational potential energy will have converted into elastic potential energy:

9.17 J = (1/2)(1860 N/m)(x²)

Step 4: Solve for Maximum Compression (x)

Now, we can solve for x, the maximum compression of the spring:

9.17 J = 930 N/m × x²

x² = 9.17 J / 930 N/m

x² ≈ 0.00985 m²

x ≈ √0.00985 m²

x ≈ 0.0992 m

Step 5: Convert Back to Centimeters

To express the compression in centimeters, we convert meters to centimeters:

x ≈ 0.0992 m × 100 = 9.92 cm

Thus, the maximum distance the spring will be compressed is approximately 9.92 cm.