Final answer:
The student inquired about the percentage change in frequency of a damped harmonic oscillator compared to its natural frequency and the number of periods to reach an amplitude of 1/e. Without specific formulas, we cannot provide a precise percentage for frequency change and an exact number of periods.
Step-by-step explanation:
The student's question relates to the damped harmonic motion of an oscillator and its frequency in comparison to its natural frequency. To find the percentage difference between the damped frequency f and the natural frequency f0, we'll need to use the decay of mechanical energy per cycle to determine the change in frequency, since the energy in a harmonic oscillator is proportional to the square of the amplitude. While we do not have the specific formula to calculate the change in frequency based on energy decay within this context, a typical approach using related physics concepts might involve complex arithmetic involving decay constants or a logarithmic relationship between amplitude and time. However, without the specific information or formulas to relate energy decay to frequency change, we cannot provide a precise percentage.
For the second part of the question, the amplitude of a damped oscillator decreases to 1/e of its original value after a number of periods. An approximation of this could be calculated by using the decay per cycle to determine the number of cycles needed for the amplitude to decrease to the given level. We would use the logarithmic formula, n = (ln(A0/A)) / (ln(1+decay percentage)), where n is the number of periods, A0 is the original amplitude, and A is the final amplitude, which is 1/e of A0. However, since we do not have a direct relation between energy decay and amplitude decay, a precise answer cannot be provided.