Energy is directly proportional to amplitude square so E1=(4)^2 divided by (8)^2 is 1 :4

Explanation:

Introduction:
In this question, we are given two identical pendulums oscillating with different amplitudes. We need to find the ratio of their energies of oscillation.

Understanding Energy of Oscillation:
The energy of oscillation of a pendulum can be divided into two components: potential energy and kinetic energy.

- Potential Energy: The potential energy of a pendulum is given by the equation PE = mgh, where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height of the bob from its equilibrium position.

- Kinetic Energy: The kinetic energy of a pendulum is given by the equation KE = (1/2)mv^2, where m is the mass of the pendulum bob and v is its velocity.

Relation between Amplitude and Energy:
In a simple harmonic motion, the amplitude of the oscillation is directly proportional to the maximum potential energy and maximum kinetic energy of the system.

- Potential Energy: The potential energy of a pendulum is directly proportional to the square of its amplitude. Therefore, if the amplitude of one pendulum is twice that of another, its potential energy will be four times that of the other pendulum.

- Kinetic Energy: The kinetic energy of a pendulum is directly proportional to the square of its amplitude. Therefore, if the amplitude of one pendulum is twice that of another, its kinetic energy will be four times that of the other pendulum.

Calculating the Ratio of Energies:
Let's assume the amplitude of the first pendulum is 4cm and the amplitude of the second pendulum is 8cm.

- First Pendulum:
The potential energy of the first pendulum is proportional to the square of its amplitude: PE1 = k * (4^2) = 16k
The kinetic energy of the first pendulum is also proportional to the square of its amplitude: KE1 = k * (4^2) = 16k

- Second Pendulum:
The potential energy of the second pendulum is proportional to the square of its amplitude: PE2 = k * (8^2) = 64k
The kinetic energy of the second pendulum is also proportional to the square of its amplitude: KE2 = k * (8^2) = 64k

- Ratio of Energies:
The ratio of the energies of oscillation of the two pendulums can be calculated by dividing the energy of the second pendulum by the energy of the first pendulum:
Ratio = (PE2 + KE2) / (PE1 + KE1)
= (64k + 64k) / (16k + 16k)
= 128k / 32k
= 4

Therefore, the ratio of the energies of oscillation of the two pendulums is 4, which corresponds to option 'B' (1/4).

Conclusion:
The ratio of the energies of oscillation of two identical pendulums is equal to the square of the ratio of their amplitudes. In this particular case, since the amplitude of the second pendulum