The image you uploaded contains a physics problem about a block of mass $M$ resting on a smooth surface, with the system involving simple harmonic motion (SHM). The problem is asking for the comparison of the frequency of oscillation of the system before and after an additional mass $m$ is placed on top of the block.

Let's break down the problem:

1. Before the mass $m$ is placed, only the block of mass $M$ is involved in the SHM.
2. After the mass $m$ is placed on top of the block, both masses $M$ and $m$ move together.

The question is asking for the ratio of frequencies after and before placing the additional mass. In SHM, the frequency is given by:

$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

Where $k$ is the spring constant and $m$ is the total effective mass involved in the oscillation.

Now, let's analyze the options:

1. Before the mass is added, the block of mass $M$ undergoes oscillation, so the frequency $f$ can be written as: $f = \frac{1}{2\pi} \sqrt{\frac{k}{M}}$

2. After the mass $m$ is added, the combined mass of $M + m$ will oscillate, so the new frequency $f'$ is: $f' = \frac{1}{2\pi} \sqrt{\frac{k}{M+m}}$

Now, the ratio of the frequencies (after and before) is:

$\frac{f'}{f} = \frac{\sqrt{\frac{k}{M+m}}}{\sqrt{\frac{k}{M}}} = \sqrt{\frac{M}{M+m}}$

Thus, the correct answer is:

$\boxed{\text{(B) } \frac{f'}{f} = \sqrt{\frac{M}{M+m}}}$

Would you like more details on how to solve this problem or any related questions?

Here are some related questions:

1. How does the mass of the object affect the frequency in SHM?
2. How would the system behave if the surface were not frictionless?
3. What happens to the frequency if you change the spring constant?
4. How is the period of oscillation related to the frequency?
5. How can you calculate the total energy of the system in SHM?

Tip: In SHM, always remember that frequency depends on the square root of the ratio of the spring constant to the total mass.