Let's break down your question step by step, starting with the force constant of the spring after it's cut into halves, and then we'll calculate the mass \( M \) based on the frequency of vibration.

Understanding the Force Constant of Each Half

When a spring is cut into two equal parts, the force constant (also known as spring constant) of each half changes. The original spring constant \( k \) is given as 3.60 N/cm, which we can convert to N/m for consistency in SI units:

• 1 N/cm = 100 N/m
• Thus, 3.60 N/cm = 3.60 × 100 = 360 N/m

Now, the relationship between the original spring constant and the spring constant of the halves can be expressed as follows:

k_half = 2 * k_original

Since each half of the spring now has double the stiffness (because the distance over which it stretches is halved), we can calculate the spring constant for each half:

• k_half = 2 * 360 N/m = 720 N/m

Calculating the Mass Supported by the Springs

Now, let's find the mass \( M \) that each spring half can support while vibrating at a frequency of 2.87 Hz. The frequency of a mass-spring system can be determined using the formula:

f = (1/2π) * √(k/m)

Rearranging this formula to solve for mass \( m \) gives us:

m = k / (4π²f²)

Substituting the values we have:

• k = 720 N/m (for each half of the spring)
• f = 2.87 Hz

Now, let’s plug in the numbers:

• m = 720 / (4π²(2.87)²)

Calculating \( 4π²(2.87)² \):

• 4π² ≈ 39.478 (using π ≈ 3.14)
• (2.87)² ≈ 8.2369
• So, 4π²(2.87)² ≈ 39.478 * 8.2369 ≈ 324.43

Now substituting back to find \( m \):

• m = 720 / 324.43 ≈ 2.22 kg

Summary of Results

To summarize:

• The force constant of each half of the spring is 720 N/m.
• The mass \( M \) that each half supports and allows the system to vibrate at 2.87 Hz is approximately 2.22 kg.

This analysis illustrates how cutting a spring impacts its properties and how those properties affect the dynamics of systems in vibration. If you have any more questions or need further clarification, feel free to ask!