The energy of a Simple Harmonic Motion (SHM) is directly proportional to the square of its frequency. In other words, if we increase the energy of the SHM by a certain factor, the frequency will also increase by the square root of that factor.

Let's assume the initial frequency of the SHM is f and the energy becomes 16 times its initial value.



The energy (E) of a SHM is given by the equation:

E = (1/2)kA^2

where k is the spring constant and A is the amplitude of the motion.

The frequency (f) of the SHM is given by the equation:

f = (1/2π)√(k/m)

where m is the mass of the object undergoing SHM.



We can express the energy in terms of frequency by substituting the value of k from the frequency equation into the energy equation:

E = (1/2)kA^2 = (1/2)(4π^2m^2f^2)A^2 = 2π^2m^2f^2A^2

Therefore, the energy (E) is directly proportional to the square of the frequency (f):

E ∝ f^2



If the energy of the SHM becomes 16 times its initial value, we can write:

16E = 2π^2m^2f^2A^2

Dividing both sides by 2π^2m^2A^2, we get:

16E/(2π^2m^2A^2) = f^2

Simplifying further, we find:

f = √(16E/(2π^2m^2A^2)) = √(8E/(π^2m^2A^2))

Therefore, the new frequency (f) is equal to the square root of 8 times the initial energy (E) divided by π^2m^2A^2.



In conclusion, if the energy of a SHM becomes sixteen times its initial value, the frequency will increase by a factor of √8.