Introduction:
In this scenario, a mass m is suspended from a spring of length l and force constant K. The frequency of vibration of the mass is f. The spring is then cut into two equal parts and the same mass is suspended from one of the parts. The new frequency of vibration of the mass is 2f. We need to determine the relationship between the frequencies.

Explanation:
To understand the relationship between the frequencies, let's analyze the situation step by step.

Step 1: Original Spring
When the mass m is suspended from the original spring, it oscillates with a certain frequency f. This frequency is determined by the mass of the object, the force constant of the spring, and the length of the spring.

Step 2: Cutting the Spring
When the spring is cut into two equal parts, the force constant K remains the same. However, the length of the spring is now reduced to half, i.e., l/2.

Step 3: New Frequency
When the same mass m is suspended from one of the parts of the cut spring, the new frequency of vibration is 2f. This means that the mass now oscillates twice as fast as it did with the original spring.

Step 4: Relationship between Frequencies
To find the relationship between the frequencies, we can use the formula for the frequency of vibration of a mass-spring system:

f = 1 / (2π) * √(K / m)

Let's denote the new frequency as f'. Since the mass m and force constant K remain the same, we can write:

f' = 1 / (2π) * √(K / m)

As per the given information, the new frequency is 2f. Substituting this into the equation, we get:

2f = 1 / (2π) * √(K / m)

Squaring both sides of the equation, we have:

(2f)^2 = (1 / (2π) * √(K / m))^2

4f^2 = (1 / (2π))^2 * (K / m)

Simplifying further, we get:

4f^2 = (1 / (4π^2)) * (K / m)

Comparing this equation with the equation for the new frequency, we can conclude that:

f' = 1 / (2π) * √(K / m) = 2f

Hence, the relationship between the frequencies is f' = 2f.

Conclusion:
When a mass is suspended from a spring, the frequency of vibration is directly proportional to the square root of the force constant and inversely proportional to the square root of the mass. When the spring is cut into two equal parts, the new frequency of vibration becomes twice the original frequency. Therefore, the relationship between the frequencies is f' = 2f.