Energy=1/2m×w^2 ×A^2
where w=angular frequency
and w=2πf
so , E= 1/2 m× 4π^2 ×f^2 × A^2
so E is directly proportional to f^2
E/16E= f^2/F' ^2
so F' = 4f is the ans.

Effect of Frequency on Energy of SHM
In a simple harmonic motion (SHM) system, the energy is directly proportional to the square of the frequency of vibration. This means that if the frequency of the SHM is increased, the energy of the system will also increase.

Initial Scenario
Let's say the SHM is vibrating with a certain frequency f. In this scenario, the energy of the SHM is at a certain level.

Energy is 16 Times
If the energy of the SHM becomes sixteen times its initial value, we need to find out the new frequency that corresponds to this energy level. Since energy is directly proportional to the square of the frequency, we can set up the following equation:
(16) = (f_new)^2 / f^2
Solving this equation, we find that the new frequency (f_new) is four times the initial frequency (f). Therefore, when the energy of the SHM becomes sixteen times its initial value, the frequency of vibration increases to four times its original value.

Explanation
When the energy of an SHM system increases, it means that the amplitude of oscillation also increases. This leads to a higher frequency of vibration as the system now covers a larger distance in the same amount of time. In other words, the system oscillates more rapidly, resulting in a higher frequency. This relationship between energy and frequency is a fundamental aspect of SHM systems.