When an object of mass m1 is hung on a vertical spring and set into vertical simple harmonic motion, it oscillates at a frequency of 14.2 Hz. When another object of mass m2 is hung on the spring along with the first object, the frequency of the motion is 3.40 Hz. Find the ratio m2/m1 of the masses.

To find the ratio of m2/m1 of the masses, we can use the relationship between the frequency (f) and mass (m) of objects undergoing vertical simple harmonic motion when connected to a spring.

The formula for the frequency of vertical simple harmonic motion is:

f = (1 / 2π) * √(k / m)

Where:
- f is the frequency of the motion
- k is the spring constant
- m is the mass of the object

Since we are considering two objects (m1 and m2) connected to the same spring, we can set up the following equation using the formula above:

For m1: f1 = (1 / 2π) * √(k / m1)
For m2: f2 = (1 / 2π) * √(k / m2)

Given that f1 = 14.2 Hz and f2 = 3.40 Hz, we can divide these two equations to eliminate k:

f1 / f2 = (1 / 2π) * √(k / m1) / (1 / 2π) * √(k / m2)

Simplifying the equation:

f1 / f2 = √(m2 / m1)

Now we can square both sides of the equation:

(f1 / f2)^2 = m2 / m1

Plugging in the given values:

(14.2 / 3.40)^2 = m2 / m1

Calculating the ratio of m2/m1:

(4.176)^2 = m2 / m1

m2 / m1 = 17.41

Therefore, the ratio of m2/m1 is approximately 17.41.