Introduction:
In this question, we are given two similar pendulums with different amplitudes of oscillations. We need to find the ratio of their energies of oscillations. To solve this problem, we will use the concept of potential and kinetic energy of a simple pendulum and apply the formulae to both pendulums.

Given:
Amplitude of the first pendulum (A1) = 2 cm
Amplitude of the second pendulum (A2) = 5 cm

Formula:
The potential energy of a simple pendulum is given by the formula:
PE = mgh

The kinetic energy of a simple pendulum is given by the formula:
KE = (1/2)mv^2

Where,
m is the mass of the pendulum bob,
g is the acceleration due to gravity,
h is the height of the bob from its mean position,
v is the velocity of the pendulum bob.

Calculations:
1. Potential Energy:
The potential energy of a pendulum bob is directly proportional to the square of its amplitude. Therefore, we can write the ratio of potential energies as:
PE1/PE2 = (A1/A2)^2

Substituting the given values, we get:
PE1/PE2 = (2/5)^2
PE1/PE2 = 4/25

2. Kinetic Energy:
The kinetic energy of a pendulum bob is directly proportional to the square of its velocity. Since the pendulums are similar, the ratio of their velocities is equal to the ratio of their amplitudes. Therefore, we can write the ratio of kinetic energies as:
KE1/KE2 = (v1/v2)^2 = (A1/A2)^2

Substituting the given values, we get:
KE1/KE2 = (2/5)^2
KE1/KE2 = 4/25

Conclusion:
The ratio of potential energies (PE1/PE2) and the ratio of kinetic energies (KE1/KE2) of the two similar pendulums are both equal to 4/25. Therefore, the ratio of their total energies of oscillations will also be 4/25.