A child plays on a bungee cord and oscillates with a certain frequency f. An adult with a mass that is 17.8 times greater than that of the child then uses the same bungee cord. What is the ratio of the frequency with the adult to the frequency with the child?

To find the ratio of the frequency with the adult to the frequency with the child, we need to understand the relationship between mass and frequency in a simple harmonic motion (SHM) system.

In SHM, the frequency (f) is inversely proportional to the square root of the mass (m). This relationship is given by the formula:

f ∝ 1/√m

where ∝ denotes "proportional to."

Now, let's denote the mass of the child as mc and the mass of the adult as ma. According to the problem, the mass of the adult (ma) is 17.8 times greater than the mass of the child (mc). Mathematically, we can express this relationship as:

ma = 17.8 * mc

Based on this information, we need to find the ratio of the frequency of the adult (fa) to the frequency of the child (fc), which can be expressed as:

fa/fc

Since the frequency is inversely proportional to the square root of the mass, we can substitute the above relationship for masses into our formula:

fa/fc = (1/√ma)/(1/√mc)
= √mc/√ma

Now, substitute the value of ma based on the given information:

fa/fc = √mc/√(17.8 * mc)
= √mc/√17.8 * √mc
= √mc/√17.8 * mc

Simplifying further:

fa/fc = 1/√17.8

Therefore, the ratio of the frequency with the adult to the frequency with the child is 1/√17.8.